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Question: Test of Hypothesis: 1. Create a word problem involving the following terms showing the different steps of hypothesis testing: a. Null Hypothesis b. Alternative Hypothesis C. Test Statistic d. Critical Region e. Critical Value f. Significance Level In the word problem, the following conditions must be satisfied: a. The mean of the sample and population are in

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Identify the null hypothesis, which is a statement that there is no effect or difference, in this context it's .

Answer). Let's say we need to test at 0.05 significance level that mean content of soft drink water is more than 12 ounces. According to the past years survey, mean conten …

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Alternative Hypothesis

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Diving deep into the realm of scientific research, the alternative hypothesis plays a pivotal role in steering investigations. It stands contrary to the null hypothesis , providing a different perspective or direction. This essential component often sets the foundation for groundbreaking discoveries. If you’re keen on understanding this concept further, our collection of alternative hypothesis statement examples, combined with a thorough writing guide and insightful tips, will serve as your comprehensive roadmap.

What is an Alternative hypothesis?

An alternative hypothesis is a statement used in statistical testing that indicates the presence of an effect, relationship, or difference. It stands in direct contrast to the null hypothesis, which posits that there is no effect or relationship. The alternative causual hypothesis provides a specific direction to the research and can be directional (e.g., one value is greater than another) or non-directional (e.g., two values are not equal).

What is an example of an Alternative hypothesis statement?

If a researcher is studying the effect of a new teaching method on student performance, the null hypothesis might be: “The new teaching method has no effect on student performance.” An example of an alternative hypothesis could be:

Directional: “Students exposed to the new teaching method will perform better than those who were not.” Non-directional: “Student performance will be different for those exposed to the new teaching method compared to those who were not.”

100 Alternative Hypothesis Statement Examples

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The alternative hypothesis symbolizes a statement of what a statistical hypothesis test is set to establish. Often contrasted with a null hypothesis, it indicates the expected direction of the tested relation. Dive into these varied thesis statement examples showcasing the core essence of alternative hypotheses.

Smoking and Cancer : Smoking is positively related to lung cancer incidence.
Diet and Weight Loss : The Atkins diet results in more weight loss than a conventional diet.
Medication Efficiency : Drug A is more effective than Drug B in treating migraines.
Exercise Duration : Engaging in physical activity for more than 30 minutes daily reduces depression symptoms.
Class Size and Learning : Smaller class sizes lead to higher student test scores.
Sugar Intake : Consuming more than 50 grams of sugar daily increases the risk of diabetes.
Vitamin C and Cold : Vitamin C intake reduces the duration of the common cold.
Sleep Duration : Sleeping less than 6 hours results in decreased cognitive function.
Training Methods : Method X training increases employee productivity more than Method Y.
Pollution Levels : Higher levels of industrial activity correlate with increased air pollution.
Stress and Disease : Chronic stress has a positive relationship with heart diseases.
Alcohol and Reaction Time : Alcohol consumption slows down reaction time.
Meditation and Blood Pressure : Regular meditation lowers blood pressure.
Organic Food : Consuming organic food leads to better gut health.
Advertising : Increased advertising results in higher sales figures.
Salary and Job Satisfaction : A higher salary correlates with job satisfaction.
Age and Memory : As age increases, short-term memory retention decreases.
Temperature and Aggression : Higher temperatures are associated with increased aggressive behavior.
Social Media : Spending more than 2 hours on social media daily increases feelings of loneliness.
Music and Concentration : Listening to classical music improves concentration during studies. …
Recycling Habits : Communities with mandatory recycling policies have higher recycling rates.
Urban Areas : Living in urban areas increases the likelihood of asthma.
Pets and Loneliness : Owning a pet decreases feelings of loneliness.
Reading Habits : Reading more than 3 books a month correlates with increased empathy.
Green Spaces : Having access to green spaces reduces stress levels.
Vaccination : Vaccination reduces the incidence of specific diseases.
Chocolate and Mood : Consuming chocolate elevates mood.
Remote Work : Working remotely improves overall work satisfaction.
Financial Literacy : Financial literacy education reduces personal debt.
Mindfulness and Anxiety : Practicing mindfulness decreases symptoms of anxiety. …
Dietary Fiber : Higher dietary fiber intake is associated with lower risks of bowel cancer.
Travel and Creativity : People who travel frequently are more creative.
Education Level and Income : Individuals with higher education levels earn more income.
Technology Adoption : People who receive technology training adapt to new devices faster.
Parental Involvement and Academic Performance : Increased parental involvement enhances students’ academic performance.
Exercise Frequency and Heart Health : Exercising at least five times a week improves heart health.
Gender and Leadership Roles : Men are more likely to hold leadership positions in corporate settings.
Social Support and Mental Health : Strong social support networks reduce the risk of depression.
Quality of Sleep and Productivity : Better sleep quality leads to higher productivity levels.
High-Fat Diet and Cholesterol Levels : A high-fat diet increases cholesterol levels.
Caffeine Intake and Alertness : Higher caffeine intake enhances alertness and cognitive function.
Online Shopping Habits : People who frequently shop online spend more money than in-store shoppers. …
Education and Political Views : Higher education levels are associated with more liberal political views.
Gender and Risk-Taking Behavior : Men are more likely to engage in risky behaviors.
Temperature and Ice Cream Sales : Higher temperatures increase ice cream sales.
Artificial Sweeteners and Weight Loss : Consuming products with artificial sweeteners aids in weight loss.
Exercise and Stress Reduction : Regular exercise reduces stress levels.
Music Genres and Mood : Listening to upbeat music improves mood.
Online Learning and Engagement : Online learners are more engaged in virtual classroom discussions.
Personality Traits and Job Performance : Extroverted individuals perform better in sales roles.
Environmental Awareness and Recycling : Higher environmental awareness leads to more recycling practices.
Social Media Usage and Self-Esteem : Excessive social media usage correlates with lower self-esteem. …
Sleep Deprivation and Reaction Time : Sleep-deprived individuals have slower reaction times.
Breakfast Consumption and Metabolism : Eating breakfast kickstarts metabolism for the day.
Leadership Style and Employee Satisfaction : Transformational leadership style increases employee job satisfaction.
Bilingualism and Cognitive Abilities : Bilingual individuals possess enhanced cognitive abilities.
Video Game Playing and Aggression : Playing violent video games increases aggressive behavior.
Hydration and Cognitive Function : Staying hydrated improves cognitive function.
Parental Support and Academic Achievement : Supportive parenting leads to higher academic achievement.
Workplace Flexibility and Work-Life Balance : Jobs with flexible schedules enhance work-life balance.
Digital Learning and Knowledge Retention : Digital learning methods improve long-term knowledge retention.
Art Exposure and Creativity : Exposure to various forms of art fosters creative thinking.
Solar Energy Adoption and Utility Bills : Homes with solar energy systems experience lower utility bills.
Parental Involvement and Student Behavior : Increased parental involvement reduces student behavioral issues.
Team Diversity and Creativity : Diverse teams generate more creative solutions.
Social Media Marketing and Brand Awareness : Social media marketing boosts brand awareness more than traditional methods.
Morning Routine and Productivity : Following a structured morning routine enhances overall productivity.
Music Training and Cognitive Development : Music training improves cognitive abilities in children.
Employee Training and Job Satisfaction : Comprehensive employee training programs lead to higher job satisfaction.
Eating Before Bed and Sleep Quality : Consuming heavy meals before bed negatively affects sleep quality.
Financial Incentives and Employee Performance : Offering financial incentives increases employee performance.
Parental Attachment and Emotional Well-being : Strong parental attachment fosters better emotional well-being in children.
Social Interaction and Mental Well-being : Frequent social interaction correlates with improved mental health.
Education and Crime Rates : Higher education levels result in lower crime rates within communities.
Diet and Acne : A diet high in dairy products exacerbates acne.
Leadership Style and Employee Motivation : Autocratic leadership style hampers employee motivation.
Urban Green Spaces and Stress Reduction : Access to urban green spaces lowers stress levels.
Sleep Duration and Athletic Performance : Adequate sleep duration enhances athletic performance.
Financial Literacy and Investment Success : Individuals with high financial literacy make more successful investments.
Team Collaboration and Project Success : Effective team collaboration leads to more successful project outcomes.
Media Exposure and Body Image : Increased media exposure contributes to negative body image perceptions.
Gender Representation and Film Success : Movies with more balanced gender representation achieve higher box office success. …
Meditation and Anxiety Reduction : Regular meditation practice reduces symptoms of anxiety.
Cognitive Training and Memory Enhancement : Cognitive training programs improve memory retention.
Positive Affirmations and Self-Confidence : Repeating positive affirmations enhances self-confidence.
Physical Fitness and Longevity : Being physically fit is linked to increased lifespan.
Parental Guidance and Online Safety : Strong parental guidance promotes responsible online behavior in children.
Artificial Intelligence and Job Displacement : Increased AI integration leads to more job displacement.
Public Transportation Usage and Air Quality : Increased public transportation usage improves air quality in cities.
Social Support and Addiction Recovery : Strong social support networks aid in addiction recovery.
Gender Diversity and Company Performance : Companies with diverse gender representation outperform others.
Mindfulness Meditation and Pain Management : Mindfulness meditation reduces perception of pain.
Music Therapy and Autism : Music therapy improves social interaction skills in children with autism.
Social Media Usage and Academic Performance : Excessive social media usage negatively impacts academic performance.
Employee Engagement and Organizational Success : Higher employee engagement leads to greater organizational success.
Healthy Eating and Longevity : A diet rich in fruits and vegetables contributes to a longer lifespan.
Gender Stereotypes and Career Choice : Gender stereotypes influence career choices among young adults.
Environmental Conservation Efforts and Biodiversity : Increased conservation efforts positively affect biodiversity.
Volunteerism and Personal Well-being : Engaging in volunteer activities enhances personal well-being.
Artificial Intelligence and Customer Service : AI-driven customer service improves user satisfaction.

Alternative Hypothesis Statement Examples in Research

In alternative research hypothesis propel investigations beyond the null. Examples span diverse fields, revealing the direction researchers expect their findings to take.

Effect of Music on Concentration : Listening to classical music enhances concentration during study.
Green Tea and Weight Loss : Green tea consumption leads to more significant weight loss than water intake.
Parental Involvement and Academic Achievement : Active parental involvement boosts student academic achievement.
Social Media Usage and Self-Esteem : Frequent social media use correlates with lower self-esteem.

Alternative Hypothesis Statement Examples in Business Research

Business research thrives on alternative hypotheses. Dive into these business-oriented examples that challenge null assumptions.

Marketing Campaign Impact : Marketing campaign A generates higher conversion rates than campaign B.
Employee Training and Productivity : Comprehensive employee training enhances workplace productivity.
Work-Life Balance and Employee Satisfaction : Improved work-life balance increases employee job satisfaction.
Customer Service Channel Effectiveness : Online chat support results in higher customer satisfaction compared to phone support.
Branding Influence on Purchase Intent : Strong brand presence leads to increased purchase intent.

Directional Alternative Hypothesis Statement Examples

Directional hypothesis add clarity to research expectations. Explore these examples that predict specific outcomes.

Exercise Frequency and Heart Health : Engaging in physical activity five times a week improves heart health.

Alternative Hypothesis Statement Examples in Psychology

Psychological studies benefit from well-crafted alternative hypotheses. These psychology hypothesis examples delve into the realm of human behavior and cognition.

Mindfulness Meditation and Anxiety Reduction : Regular mindfulness practice reduces symptoms of anxiety.

Alternative Null Hypothesis Statement Examples

Explore alternative null hypothesis —statements asserting the absence of specific effects or differences.

Coffee Consumption and Weight Gain : Increased coffee consumption does not lead to weight gain.
Smartphone Usage and Sleep Quality : Using smartphones before bed does not impact sleep quality.
Music Genre and Study Performance : Studying with rock music does not affect academic performance.
Green Spaces and Stress Reduction : Access to green spaces does not decrease stress levels.
Team Diversity and Project Success : Team diversity does not influence project success rates.

Alternative Hypothesis Statement Examples in Medical Research

Medical research relies on robust alternative hypotheses to drive scientific inquiry. These examples explore hypotheses in the realm of healthcare.

Exercise and Diabetes Prevention : Regular exercise decreases the risk of developing type 2 diabetes.
Medication A and Blood Pressure Reduction : Medication A leads to greater reduction in blood pressure compared to medication B.
Nutritional Intake and Heart Disease : Higher intake of fruits and vegetables lowers the risk of heart disease.
Stress Reduction Techniques and Anxiety Levels : Practicing stress reduction techniques decreases anxiety levels.
Alternative Medicine and Pain Management : Alternative medicine therapies alleviate chronic pain more effectively than traditional treatments.

Alternative Hypothesis Statement Examples in Education Research

Education research thrives on alternative hypotheses to investigate innovative approaches. Explore examples that challenge conventional notions.

Technology Integration and Student Engagement : Integrating technology enhances student engagement in the classroom.
Project-Based Learning and Knowledge Retention : Project-based learning improves long-term knowledge retention.
Teacher Professional Development and Student Performance : Effective teacher professional development positively impacts student academic performance.
Inclusive Classroom Environment and Learning Outcomes : Inclusive classrooms lead to better learning outcomes for diverse students.
Feedback Frequency and Writing Improvement : Frequent feedback results in greater improvement in student writing skills.

These examples showcase the pivotal role of alternative hypotheses across various disciplines, serving as the driving force behind scientific exploration and advancement.

What is the Alternative Hypothesis Formula?

The alternative hypothesis, denoted as “Ha” or “H1,” represents the assertion researchers aim to support through evidence. It stands in contrast to the null hypothesis (Ho), which suggests no effect or relationship. The formula for the alternative hypothesis varies based on the nature of the study:

Directional Hypothesis : For studies with an expected direction, the formula takes the form of a prediction. For instance, “The new drug increases patient recovery rates.”
Non-Directional Hypothesis : For exploratory studies, the formula reflects the possibility of any difference or effect. For example, “There is a difference in recovery rates between the two drugs.”

How do you start an Alternative Hypothesis?

Starting an alternative simple hypothesis involves framing a clear research statement that highlights the anticipated effect, relationship, or difference. To begin:

Identify the Research Question: Determine the specific aspect you intend to explore or compare.
Formulate a Hypothesis: Craft a statement that directly addresses the expected outcome.
Include Variables: Introduce the relevant variables and their predicted connection.
Be Clear and Specific: Ensure the alternative hypothesis is concise and unambiguous.

Is the Alternative Hypothesis a Claim or Statement?

The alternative hypothesis is both a claim and a statement. It claims that there is a measurable effect, relationship, or difference in the variables being studied. It is also a statement that researchers work to validate through evidence.

How do you write an Alternative Hypothesis Statement? – Step by Step Guide

Creating a robust alternative hypothesis statement involves structured steps:

Identify Variables : Clearly define the independent and dependent variables in your study.
State Expected Effect : Express the anticipated impact, relationship, or difference between variables.
Be Precise : Use specific language to convey the exact nature of the expected outcome.
Include Direction (if applicable) : If your hypothesis is directional, specify the expected direction.
Avoid Ambiguity : Make sure your statement is clear and leaves no room for confusion.

Tips for Writing an Alternative Hypothesis Statement

Be Specific : Clearly define the variables and the predicted relationship.
Use Measurable Terms : Incorporate quantifiable terms to indicate the magnitude of the effect.
Testability : Ensure the hypothesis can be tested empirically.
Conciseness : Keep the statement concise and to the point.
Alignment with Research Question : Ensure the hypothesis directly answers your research question.
Avoid Value Judgments : Avoid value judgments or personal biases in the hypothesis.
Review Literature : Consult existing literature to align your hypothesis with prior research.

Crafting a strong alternative hypothesis statement is essential for guiding your research and forming the basis for causual hypothesis testing. It directs the focus of your investigation and lays the foundation for drawing meaningful conclusions.

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9.E: Hypothesis Testing with One Sample (Exercises)

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These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

9.1: Introduction

9.2: null and alternative hypotheses.

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, $H_{0}$, and the alternative hypothesis. $H_{a}$, in terms of the appropriate parameter $(\mu \text{or} p)$.

The mean number of years Americans work before retiring is 34.
At most 60% of Americans vote in presidential elections.
The mean starting salary for San Jose State University graduates is at least $100,000 per year.
Twenty-nine percent of high school seniors get drunk each month.
Fewer than 5% of adults ride the bus to work in Los Angeles.
The mean number of cars a person owns in her lifetime is not more than ten.
About half of Americans prefer to live away from cities, given the choice.
Europeans have a mean paid vacation each year of six weeks.
The chance of developing breast cancer is under 11% for women.
Private universities' mean tuition cost is more than $20,000 per year.
$H_{0}: \mu = 34; H_{a}: \mu \neq 34$
$H_{0}: p \leq 0.60; H_{a}: p > 0.60$
$H_{0}: \mu \geq 100,000; H_{a}: \mu < 100,000$
$H_{0}: p = 0.29; H_{a}: p \neq 0.29$
$H_{0}: p = 0.05; H_{a}: p < 0.05$
$H_{0}: \mu \leq 10; H_{a}: \mu > 10$
$H_{0}: p = 0.50; H_{a}: p \neq 0.50$
$H_{0}: \mu = 6; H_{a}: \mu \neq 6$
$H_{0}: p ≥ 0.11; H_{a}: p < 0.11$
$H_{0}: \mu \leq 20,000; H_{a}: \mu > 20,000$

Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:

$p < 0.30$
$p \leq 0.30$
$p \geq 0.30$
$p > 0.30$

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:

$p = 0.20$
$p > 0.20$
$p < 0.20$
$p \leq 0.20$

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:

$H_{0}: \bar{x} = 4.5, H_{a}: \bar{x} > 4.5$
$H_{0}: \mu \geq 4.5, H_{a}: \mu < 4.5$
$H_{0}: \mu = 4.75, H_{a}: \mu > 4.75$
$H_{0}: \mu = 4.5, H_{a}: \mu > 4.5$

9.3: Outcomes and the Type I and Type II Errors

State the Type I and Type II errors in complete sentences given the following statements.

The mean number of cars a person owns in his or her lifetime is not more than ten.
Private universities mean tuition cost is more than $20,000 per year.
Type I error: We conclude that the mean is not 34 years, when it really is 34 years. Type II error: We conclude that the mean is 34 years, when in fact it really is not 34 years.
Type I error: We conclude that more than 60% of Americans vote in presidential elections, when the actual percentage is at most 60%.Type II error: We conclude that at most 60% of Americans vote in presidential elections when, in fact, more than 60% do.
Type I error: We conclude that the mean starting salary is less than $100,000, when it really is at least $100,000. Type II error: We conclude that the mean starting salary is at least $100,000 when, in fact, it is less than $100,000.
Type I error: We conclude that the proportion of high school seniors who get drunk each month is not 29%, when it really is 29%. Type II error: We conclude that the proportion of high school seniors who get drunk each month is 29% when, in fact, it is not 29%.
Type I error: We conclude that fewer than 5% of adults ride the bus to work in Los Angeles, when the percentage that do is really 5% or more. Type II error: We conclude that 5% or more adults ride the bus to work in Los Angeles when, in fact, fewer that 5% do.
Type I error: We conclude that the mean number of cars a person owns in his or her lifetime is more than 10, when in reality it is not more than 10. Type II error: We conclude that the mean number of cars a person owns in his or her lifetime is not more than 10 when, in fact, it is more than 10.
Type I error: We conclude that the proportion of Americans who prefer to live away from cities is not about half, though the actual proportion is about half. Type II error: We conclude that the proportion of Americans who prefer to live away from cities is half when, in fact, it is not half.
Type I error: We conclude that the duration of paid vacations each year for Europeans is not six weeks, when in fact it is six weeks. Type II error: We conclude that the duration of paid vacations each year for Europeans is six weeks when, in fact, it is not.
Type I error: We conclude that the proportion is less than 11%, when it is really at least 11%. Type II error: We conclude that the proportion of women who develop breast cancer is at least 11%, when in fact it is less than 11%.
Type I error: We conclude that the average tuition cost at private universities is more than $20,000, though in reality it is at most $20,000. Type II error: We conclude that the average tuition cost at private universities is at most $20,000 when, in fact, it is more than $20,000.

For statements a-j in Exercise 9.109 , answer the following in complete sentences.

State a consequence of committing a Type I error.
State a consequence of committing a Type II error.

When a new drug is created, the pharmaceutical company must subject it to testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug. Suppose the null hypothesis is “the drug is unsafe.” What is the Type II Error?

To conclude the drug is safe when in, fact, it is unsafe.
Not to conclude the drug is safe when, in fact, it is safe.
To conclude the drug is safe when, in fact, it is safe.
Not to conclude the drug is unsafe when, in fact, it is unsafe.

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 of them attended the midnight showing. The Type I error is to conclude that the percent of EVC students who attended is ________.

at least 20%, when in fact, it is less than 20%.
20%, when in fact, it is 20%.
less than 20%, when in fact, it is at least 20%.
less than 20%, when in fact, it is less than 20%.

It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average?

The Type II error is not to reject that the mean number of hours of sleep LTCC students get per night is at least seven when, in fact, the mean number of hours

is more than seven hours.
is at most seven hours.
is at least seven hours.
is less than seven hours.

to conclude that the current mean hours per week is higher than 4.5, when in fact, it is higher
to conclude that the current mean hours per week is higher than 4.5, when in fact, it is the same
to conclude that the mean hours per week currently is 4.5, when in fact, it is higher
to conclude that the mean hours per week currently is no higher than 4.5, when in fact, it is not higher

9.4: Distribution Needed for Hypothesis Testing

$N\left(7.24, \frac{1.93}{\sqrt{22}}\right)$
$N\left(7.24, 1.93\right)$

9.5: Rare Events, the Sample, Decision and Conclusion

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. Conduct a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population.

Is this a test of one mean or proportion?
State the null and alternative hypotheses. $H_{0}$ : ____________________ $H_{a}$ : ____________________
Is this a right-tailed, left-tailed, or two-tailed test?
What symbol represents the random variable for this test?
In words, define the random variable for this test.
$x =$ ________________
$n =$ ________________
$p′ =$ _____________
Calculate $\sigma_{x} =$ __________. Show the formula set-up.
State the distribution to use for the hypothesis test.
Find the $p\text{-value}$.
Reason for the decision:
Conclusion (write out in a complete sentence):

9.6: Additional Information and Full Hypothesis Test Examples

For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in [link] . Please feel free to make copies of the solution sheets. For the online version of the book, it is suggested that you copy the .doc or the .pdf files.

If you are using a Student's $t$ - distribution for one of the following homework problems, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, however.)

A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles. Using $\alpha = 0.05$, is the data highly inconsistent with the claim?

$H_{0}: \mu \geq 50,000$
$H_{a}: \mu < 50,000$
Let $\bar{X} =$ the average lifespan of a brand of tires.
normal distribution
$z = -2.315$
$p\text{-value} = 0.0103$
Check student’s solution.
alpha: 0.05
Decision: Reject the null hypothesis.
Reason for decision: The $p\text{-value}$ is less than 0.05.
Conclusion: There is sufficient evidence to conclude that the mean lifespan of the tires is less than 50,000 miles.
$(43,537, 49,463)$

From generation to generation, the mean age when smokers first start to smoke varies. However, the standard deviation of that age remains constant of around 2.1 years. A survey of 40 smokers of this generation was done to see if the mean starting age is at least 19. The sample mean was 18.1 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level?

The cost of a daily newspaper varies from city to city. However, the variation among prices remains steady with a standard deviation of 20¢. A study was done to test the claim that the mean cost of a daily newspaper is $1.00. Twelve costs yield a mean cost of 95¢ with a standard deviation of 18¢. Do the data support the claim at the 1% level?

$H_{0}: \mu = $1.00$
$H_{a}: \mu \neq $1.00$
Let $\bar{X} =$ the average cost of a daily newspaper.
$z = –0.866$
$p\text{-value} = 0.3865$
$\alpha: 0.01$
Decision: Do not reject the null hypothesis.
Reason for decision: The $p\text{-value}$ is greater than 0.01.
Conclusion: There is sufficient evidence to support the claim that the mean cost of daily papers is $1. The mean cost could be $1.
$($0.84, $1.06)$

An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% level?

The mean number of sick days an employee takes per year is believed to be about ten. Members of a personnel department do not believe this figure. They randomly survey eight employees. The number of sick days they took for the past year are as follows: 12; 4; 15; 3; 11; 8; 6; 8. Let $x =$ the number of sick days they took for the past year. Should the personnel team believe that the mean number is ten?

$H_{0}: \mu = 10$
$H_{a}: \mu \neq 10$
Let $\bar{X}$ the mean number of sick days an employee takes per year.
Student’s t -distribution
$t = –1.12$
$p\text{-value} = 0.300$
$\alpha: 0.05$
Reason for decision: The $p\text{-value}$ is greater than 0.05.
Conclusion: At the 5% significance level, there is insufficient evidence to conclude that the mean number of sick days is not ten.
$(4.9443, 11.806)$

In 1955, Life Magazine reported that the 25 year-old mother of three worked, on average, an 80 hour week. Recently, many groups have been studying whether or not the women's movement has, in fact, resulted in an increase in the average work week for women (combining employment and at-home work). Suppose a study was done to determine if the mean work week has increased. 81 women were surveyed with the following results. The sample mean was 83; the sample standard deviation was ten. Does it appear that the mean work week has increased for women at the 5% level?

Your statistics instructor claims that 60 percent of the students who take her Elementary Statistics class go through life feeling more enriched. For some reason that she can't quite figure out, most people don't believe her. You decide to check this out on your own. You randomly survey 64 of her past Elementary Statistics students and find that 34 feel more enriched as a result of her class. Now, what do you think?

$H_{0}: p \geq 0.6$
$H_{a}: p < 0.6$
Let $P′ =$ the proportion of students who feel more enriched as a result of taking Elementary Statistics.
normal for a single proportion
$p\text{-value} = 0.1308$
Conclusion: There is insufficient evidence to conclude that less than 60 percent of her students feel more enriched.

The “plus-4s” confidence interval is $(0.411, 0.648)$

A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.

Refer to Exercise 9.119 . Conduct a hypothesis test to see if your decision and conclusion would change if your belief were that the brown trout’s mean I.Q. is not four.

$H_{0}: \mu = 4$
$H_{a}: \mu \neq 4$
Let $\bar{X}$ the average I.Q. of a set of brown trout.
two-tailed Student's t-test
$t = 1.95$
$p\text{-value} = 0.076$
Reason for decision: The $p\text{-value}$ is greater than 0.05
Conclusion: There is insufficient evidence to conclude that the average IQ of brown trout is not four.
$(3.8865,5.9468)$

According to an article in Newsweek , the natural ratio of girls to boys is 100:105. In China, the birth ratio is 100: 114 (46.7% girls). Suppose you don’t believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 60 girls and 90 boys born of the 150. Based on your study, do you believe that the percent of girls born in China is 46.7?

A poll done for Newsweek found that 13% of Americans have seen or sensed the presence of an angel. A contingent doubts that the percent is really that high. It conducts its own survey. Out of 76 Americans surveyed, only two had seen or sensed the presence of an angel. As a result of the contingent’s survey, would you agree with the Newsweek poll? In complete sentences, also give three reasons why the two polls might give different results.

$H_{a}: p < 0.13$
Let $P′ =$ the proportion of Americans who have seen or sensed angels
–2.688
$p\text{-value} = 0.0036$
Reason for decision: The $p\text{-value}$e is less than 0.05.
Conclusion: There is sufficient evidence to conclude that the percentage of Americans who have seen or sensed an angel is less than 13%.

The“plus-4s” confidence interval is (0.0022, 0.0978)

The mean work week for engineers in a start-up company is believed to be about 60 hours. A newly hired engineer hopes that it’s shorter. She asks ten engineering friends in start-ups for the lengths of their mean work weeks. Based on the results that follow, should she count on the mean work week to be shorter than 60 hours?

Data (length of mean work week): 70; 45; 55; 60; 65; 55; 55; 60; 50; 55.

Use the “Lap time” data for Lap 4 (see [link] ) to test the claim that Terri finishes Lap 4, on average, in less than 129 seconds. Use all twenty races given.

$H_{0}: \mu \geq 129$
$H_{a}: \mu < 129$
Let $\bar{X} =$ the average time in seconds that Terri finishes Lap 4.
Student's t -distribution
$t = 1.209$
Conclusion: There is insufficient evidence to conclude that Terri’s mean lap time is less than 129 seconds.
$(128.63, 130.37)$

Use the “Initial Public Offering” data (see [link] ) to test the claim that the mean offer price was $18 per share. Do not use all the data. Use your random number generator to randomly survey 15 prices.

The following questions were written by past students. They are excellent problems!

"Asian Family Reunion," by Chau Nguyen

Every two years it comes around.

We all get together from different towns.

In my honest opinion,

It's not a typical family reunion.

Not forty, or fifty, or sixty,

But how about seventy companions!

The kids would play, scream, and shout

One minute they're happy, another they'll pout.

The teenagers would look, stare, and compare

From how they look to what they wear.

The men would chat about their business

That they make more, but never less.

Money is always their subject

And there's always talk of more new projects.

The women get tired from all of the chats

They head to the kitchen to set out the mats.

Some would sit and some would stand

Eating and talking with plates in their hands.

Then come the games and the songs

And suddenly, everyone gets along!

With all that laughter, it's sad to say

That it always ends in the same old way.

They hug and kiss and say "good-bye"

And then they all begin to cry!

I say that 60 percent shed their tears

But my mom counted 35 people this year.

She said that boys and men will always have their pride,

So we won't ever see them cry.

I myself don't think she's correct,

So could you please try this problem to see if you object?

$H_{0}: p = 0.60$
$H_{a}: p < 0.60$
Let $P′ =$ the proportion of family members who shed tears at a reunion.
–1.71
Reason for decision: $p\text{-value} < \alpha$
Conclusion: At the 5% significance level, there is sufficient evidence to conclude that the proportion of family members who shed tears at a reunion is less than 0.60. However, the test is weak because the $p\text{-value}$ and alpha are quite close, so other tests should be done.
We are 95% confident that between 38.29% and 61.71% of family members will shed tears at a family reunion. $(0.3829, 0.6171)$. The“plus-4s” confidence interval (see chapter 8) is $(0.3861, 0.6139)$

Note that here the “large-sample” $1 - \text{PropZTest}$ provides the approximate $p\text{-value}$ of 0.0438. Whenever a $p\text{-value}$ based on a normal approximation is close to the level of significance, the exact $p\text{-value}$ based on binomial probabilities should be calculated whenever possible. This is beyond the scope of this course.

"The Problem with Angels," by Cyndy Dowling

Although this problem is wholly mine,

The catalyst came from the magazine, Time.

On the magazine cover I did find

The realm of angels tickling my mind.

Inside, 69% I found to be

In angels, Americans do believe.

Then, it was time to rise to the task,

Ninety-five high school and college students I did ask.

Viewing all as one group,

Random sampling to get the scoop.

So, I asked each to be true,

"Do you believe in angels?" Tell me, do!

Hypothesizing at the start,

Totally believing in my heart

That the proportion who said yes

Would be equal on this test.

Lo and behold, seventy-three did arrive,

Out of the sample of ninety-five.

Now your job has just begun,

Solve this problem and have some fun.

"Blowing Bubbles," by Sondra Prull

Studying stats just made me tense,

I had to find some sane defense.

Some light and lifting simple play

To float my math anxiety away.

Blowing bubbles lifts me high

Takes my troubles to the sky.

POIK! They're gone, with all my stress

Bubble therapy is the best.

The label said each time I blew

The average number of bubbles would be at least 22.

I blew and blew and this I found

From 64 blows, they all are round!

But the number of bubbles in 64 blows

Varied widely, this I know.

20 per blow became the mean

They deviated by 6, and not 16.

From counting bubbles, I sure did relax

But now I give to you your task.

Was 22 a reasonable guess?

Find the answer and pass this test!

$H_{0}: \mu \geq 22$
$H_{a}: \mu < 22$
Let $\bar{X} =$ the mean number of bubbles per blow.
–2.667
$p\text{-value} = 0.00486$
Conclusion: There is sufficient evidence to conclude that the mean number of bubbles per blow is less than 22.
$(18.501, 21.499)$

"Dalmatian Darnation," by Kathy Sparling

A greedy dog breeder named Spreckles

Bred puppies with numerous freckles

The Dalmatians he sought

Possessed spot upon spot

The more spots, he thought, the more shekels.

His competitors did not agree

That freckles would increase the fee.

They said, “Spots are quite nice

But they don't affect price;

One should breed for improved pedigree.”

The breeders decided to prove

This strategy was a wrong move.

Breeding only for spots

Would wreak havoc, they thought.

His theory they want to disprove.

They proposed a contest to Spreckles

Comparing dog prices to freckles.

In records they looked up

One hundred one pups:

Dalmatians that fetched the most shekels.

They asked Mr. Spreckles to name

An average spot count he'd claim

To bring in big bucks.

Said Spreckles, “Well, shucks,

It's for one hundred one that I aim.”

Said an amateur statistician

Who wanted to help with this mission.

“Twenty-one for the sample

Standard deviation's ample:

They examined one hundred and one

Dalmatians that fetched a good sum.

They counted each spot,

Mark, freckle and dot

And tallied up every one.

Instead of one hundred one spots

They averaged ninety six dots

Can they muzzle Spreckles’

Obsession with freckles

Based on all the dog data they've got?

"Macaroni and Cheese, please!!" by Nedda Misherghi and Rachelle Hall

As a poor starving student I don't have much money to spend for even the bare necessities. So my favorite and main staple food is macaroni and cheese. It's high in taste and low in cost and nutritional value.

One day, as I sat down to determine the meaning of life, I got a serious craving for this, oh, so important, food of my life. So I went down the street to Greatway to get a box of macaroni and cheese, but it was SO expensive! $2.02 !!! Can you believe it? It made me stop and think. The world is changing fast. I had thought that the mean cost of a box (the normal size, not some super-gigantic-family-value-pack) was at most $1, but now I wasn't so sure. However, I was determined to find out. I went to 53 of the closest grocery stores and surveyed the prices of macaroni and cheese. Here are the data I wrote in my notebook:

Price per box of Mac and Cheese:

5 stores @ $2.02
15 stores @ $0.25
3 stores @ $1.29
6 stores @ $0.35
4 stores @ $2.27
7 stores @ $1.50
5 stores @ $1.89
8 stores @ 0.75.

I could see that the cost varied but I had to sit down to figure out whether or not I was right. If it does turn out that this mouth-watering dish is at most $1, then I'll throw a big cheesy party in our next statistics lab, with enough macaroni and cheese for just me. (After all, as a poor starving student I can't be expected to feed our class of animals!)

$H_{0}: \mu \leq 1$
$H_{a}: \mu > 1$
Let $\bar{X} =$ the mean cost in dollars of macaroni and cheese in a certain town.
Student's $t$-distribution
$t = 0.340$
$p\text{-value} = 0.36756$
Conclusion: The mean cost could be $1, or less. At the 5% significance level, there is insufficient evidence to conclude that the mean price of a box of macaroni and cheese is more than $1.
$(0.8291, 1.241)$

"William Shakespeare: The Tragedy of Hamlet, Prince of Denmark," by Jacqueline Ghodsi

THE CHARACTERS (in order of appearance):

HAMLET, Prince of Denmark and student of Statistics
POLONIUS, Hamlet’s tutor
HOROTIO, friend to Hamlet and fellow student

Scene: The great library of the castle, in which Hamlet does his lessons

(The day is fair, but the face of Hamlet is clouded. He paces the large room. His tutor, Polonius, is reprimanding Hamlet regarding the latter’s recent experience. Horatio is seated at the large table at right stage.)

POLONIUS: My Lord, how cans’t thou admit that thou hast seen a ghost! It is but a figment of your imagination!

HAMLET: I beg to differ; I know of a certainty that five-and-seventy in one hundred of us, condemned to the whips and scorns of time as we are, have gazed upon a spirit of health, or goblin damn’d, be their intents wicked or charitable.

POLONIUS If thou doest insist upon thy wretched vision then let me invest your time; be true to thy work and speak to me through the reason of the null and alternate hypotheses. (He turns to Horatio.) Did not Hamlet himself say, “What piece of work is man, how noble in reason, how infinite in faculties? Then let not this foolishness persist. Go, Horatio, make a survey of three-and-sixty and discover what the true proportion be. For my part, I will never succumb to this fantasy, but deem man to be devoid of all reason should thy proposal of at least five-and-seventy in one hundred hold true.

HORATIO (to Hamlet): What should we do, my Lord?

HAMLET: Go to thy purpose, Horatio.

HORATIO: To what end, my Lord?

HAMLET: That you must teach me. But let me conjure you by the rights of our fellowship, by the consonance of our youth, but the obligation of our ever-preserved love, be even and direct with me, whether I am right or no.

(Horatio exits, followed by Polonius, leaving Hamlet to ponder alone.)

(The next day, Hamlet awaits anxiously the presence of his friend, Horatio. Polonius enters and places some books upon the table just a moment before Horatio enters.)

POLONIUS: So, Horatio, what is it thou didst reveal through thy deliberations?

HORATIO: In a random survey, for which purpose thou thyself sent me forth, I did discover that one-and-forty believe fervently that the spirits of the dead walk with us. Before my God, I might not this believe, without the sensible and true avouch of mine own eyes.

POLONIUS: Give thine own thoughts no tongue, Horatio. (Polonius turns to Hamlet.) But look to’t I charge you, my Lord. Come Horatio, let us go together, for this is not our test. (Horatio and Polonius leave together.)

HAMLET: To reject, or not reject, that is the question: whether ‘tis nobler in the mind to suffer the slings and arrows of outrageous statistics, or to take arms against a sea of data, and, by opposing, end them. (Hamlet resignedly attends to his task.)

(Curtain falls)

"Untitled," by Stephen Chen

I've often wondered how software is released and sold to the public. Ironically, I work for a company that sells products with known problems. Unfortunately, most of the problems are difficult to create, which makes them difficult to fix. I usually use the test program X, which tests the product, to try to create a specific problem. When the test program is run to make an error occur, the likelihood of generating an error is 1%.

So, armed with this knowledge, I wrote a new test program Y that will generate the same error that test program X creates, but more often. To find out if my test program is better than the original, so that I can convince the management that I'm right, I ran my test program to find out how often I can generate the same error. When I ran my test program 50 times, I generated the error twice. While this may not seem much better, I think that I can convince the management to use my test program instead of the original test program. Am I right?

$H_{0}: p = 0.01$
$H_{a}: p > 0.01$
Let $P′ =$ the proportion of errors generated
Normal for a single proportion
Decision: Reject the null hypothesis
Conclusion: At the 5% significance level, there is sufficient evidence to conclude that the proportion of errors generated is more than 0.01.

The“plus-4s” confidence interval is $(0.004, 0.144)$.

"Japanese Girls’ Names"

by Kumi Furuichi

It used to be very typical for Japanese girls’ names to end with “ko.” (The trend might have started around my grandmothers’ generation and its peak might have been around my mother’s generation.) “Ko” means “child” in Chinese characters. Parents would name their daughters with “ko” attaching to other Chinese characters which have meanings that they want their daughters to become, such as Sachiko—happy child, Yoshiko—a good child, Yasuko—a healthy child, and so on.

However, I noticed recently that only two out of nine of my Japanese girlfriends at this school have names which end with “ko.” More and more, parents seem to have become creative, modernized, and, sometimes, westernized in naming their children.

I have a feeling that, while 70 percent or more of my mother’s generation would have names with “ko” at the end, the proportion has dropped among my peers. I wrote down all my Japanese friends’, ex-classmates’, co-workers, and acquaintances’ names that I could remember. Following are the names. (Some are repeats.) Test to see if the proportion has dropped for this generation.

Ai, Akemi, Akiko, Ayumi, Chiaki, Chie, Eiko, Eri, Eriko, Fumiko, Harumi, Hitomi, Hiroko, Hiroko, Hidemi, Hisako, Hinako, Izumi, Izumi, Junko, Junko, Kana, Kanako, Kanayo, Kayo, Kayoko, Kazumi, Keiko, Keiko, Kei, Kumi, Kumiko, Kyoko, Kyoko, Madoka, Maho, Mai, Maiko, Maki, Miki, Miki, Mikiko, Mina, Minako, Miyako, Momoko, Nana, Naoko, Naoko, Naoko, Noriko, Rieko, Rika, Rika, Rumiko, Rei, Reiko, Reiko, Sachiko, Sachiko, Sachiyo, Saki, Sayaka, Sayoko, Sayuri, Seiko, Shiho, Shizuka, Sumiko, Takako, Takako, Tomoe, Tomoe, Tomoko, Touko, Yasuko, Yasuko, Yasuyo, Yoko, Yoko, Yoko, Yoshiko, Yoshiko, Yoshiko, Yuka, Yuki, Yuki, Yukiko, Yuko, Yuko.

"Phillip’s Wish," by Suzanne Osorio

My nephew likes to play

Chasing the girls makes his day.

He asked his mother

If it is okay

To get his ear pierced.

She said, “No way!”

To poke a hole through your ear,

Is not what I want for you, dear.

He argued his point quite well,

Says even my macho pal, Mel,

Has gotten this done.

It’s all just for fun.

C’mon please, mom, please, what the hell.

Again Phillip complained to his mother,

Saying half his friends (including their brothers)

Are piercing their ears

And they have no fears

He wants to be like the others.

She said, “I think it’s much less.

We must do a hypothesis test.

And if you are right,

I won’t put up a fight.

But, if not, then my case will rest.”

We proceeded to call fifty guys

To see whose prediction would fly.

Nineteen of the fifty

Said piercing was nifty

And earrings they’d occasionally buy.

Then there’s the other thirty-one,

Who said they’d never have this done.

So now this poem’s finished.

Will his hopes be diminished,

Or will my nephew have his fun?

$H_{0}: p = 0.50$
$H_{a}: p < 0.50$
Let $P′ =$ the proportion of friends that has a pierced ear.
–1.70
$p\text{-value} = 0.0448$
Reason for decision: The $p\text{-value}$ is less than 0.05. (However, they are very close.)
Conclusion: There is sufficient evidence to support the claim that less than 50% of his friends have pierced ears.
Confidence Interval: $(0.245, 0.515)$: The “plus-4s” confidence interval is $(0.259, 0.519)$.

"The Craven," by Mark Salangsang

Once upon a morning dreary

In stats class I was weak and weary.

Pondering over last night’s homework

Whose answers were now on the board

This I did and nothing more.

While I nodded nearly napping

Suddenly, there came a tapping.

As someone gently rapping,

Rapping my head as I snore.

Quoth the teacher, “Sleep no more.”

“In every class you fall asleep,”

The teacher said, his voice was deep.

“So a tally I’ve begun to keep

Of every class you nap and snore.

The percentage being forty-four.”

“My dear teacher I must confess,

While sleeping is what I do best.

The percentage, I think, must be less,

A percentage less than forty-four.”

This I said and nothing more.

“We’ll see,” he said and walked away,

And fifty classes from that day

He counted till the month of May

The classes in which I napped and snored.

The number he found was twenty-four.

At a significance level of 0.05,

Please tell me am I still alive?

Or did my grade just take a dive

Plunging down beneath the floor?

Upon thee I hereby implore.

Toastmasters International cites a report by Gallop Poll that 40% of Americans fear public speaking. A student believes that less than 40% of students at her school fear public speaking. She randomly surveys 361 schoolmates and finds that 135 report they fear public speaking. Conduct a hypothesis test to determine if the percent at her school is less than 40%.

$H_{0}: p = 0.40$
$H_{a}: p < 0.40$
Let $P′ =$ the proportion of schoolmates who fear public speaking.
–1.01
$p\text{-value} = 0.1563$
Conclusion: There is insufficient evidence to support the claim that less than 40% of students at the school fear public speaking.
Confidence Interval: $(0.3241, 0.4240)$: The “plus-4s” confidence interval is $(0.3257, 0.4250)$.

Sixty-eight percent of online courses taught at community colleges nationwide were taught by full-time faculty. To test if 68% also represents California’s percent for full-time faculty teaching the online classes, Long Beach City College (LBCC) in California, was randomly selected for comparison. In the same year, 34 of the 44 online courses LBCC offered were taught by full-time faculty. Conduct a hypothesis test to determine if 68% represents California. NOTE: For more accurate results, use more California community colleges and this past year's data.

According to an article in Bloomberg Businessweek , New York City's most recent adult smoking rate is 14%. Suppose that a survey is conducted to determine this year’s rate. Nine out of 70 randomly chosen N.Y. City residents reply that they smoke. Conduct a hypothesis test to determine if the rate is still 14% or if it has decreased.

$H_{0}: p = 0.14$
$H_{a}: p < 0.14$
Let $P′ =$ the proportion of NYC residents that smoke.
–0.2756
$p\text{-value} = 0.3914$
At the 5% significance level, there is insufficient evidence to conclude that the proportion of NYC residents who smoke is less than 0.14.
Confidence Interval: $(0.0502, 0.2070)$: The “plus-4s” confidence interval (see chapter 8) is $(0.0676, 0.2297)$.

The mean age of De Anza College students in a previous term was 26.6 years old. An instructor thinks the mean age for online students is older than 26.6. She randomly surveys 56 online students and finds that the sample mean is 29.4 with a standard deviation of 2.1. Conduct a hypothesis test.

Registered nurses earned an average annual salary of $69,110. For that same year, a survey was conducted of 41 California registered nurses to determine if the annual salary is higher than $69,110 for California nurses. The sample average was $71,121 with a sample standard deviation of $7,489. Conduct a hypothesis test.

$H_{0}: \mu = 69,110$
$H_{0}: \mu > 69,110$
Let $\bar{X} =$ the mean salary in dollars for California registered nurses.
$t = 1.719$
$p\text{-value}: 0.0466$
Conclusion: At the 5% significance level, there is sufficient evidence to conclude that the mean salary of California registered nurses exceeds $69,110.
$($68,757, $73,485)$

La Leche League International reports that the mean age of weaning a child from breastfeeding is age four to five worldwide. In America, most nursing mothers wean their children much earlier. Suppose a random survey is conducted of 21 U.S. mothers who recently weaned their children. The mean weaning age was nine months (3/4 year) with a standard deviation of 4 months. Conduct a hypothesis test to determine if the mean weaning age in the U.S. is less than four years old.

After conducting the test, your decision and conclusion are

Reject $H_{0}$: There is sufficient evidence to conclude that more than 30% of teen girls smoke to stay thin.
Do not reject $H_{0}$: There is not sufficient evidence to conclude that less than 30% of teen girls smoke to stay thin.
Do not reject $H_{0}$: There is not sufficient evidence to conclude that more than 30% of teen girls smoke to stay thin.
Reject $H_{0}$: There is sufficient evidence to conclude that less than 30% of teen girls smoke to stay thin.

At a 1% level of significance, an appropriate conclusion is:

There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than 20%.
There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is more than 20%.
There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than 20%.
There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is at least 20%.

At a significance level of $a = 0.05$, what is the correct conclusion?

There is enough evidence to conclude that the mean number of hours is more than 4.75
There is enough evidence to conclude that the mean number of hours is more than 4.5
There is not enough evidence to conclude that the mean number of hours is more than 4.5
There is not enough evidence to conclude that the mean number of hours is more than 4.75

Instructions: For the following ten exercises,

Hypothesis testing: For the following ten exercises, answer each question.

State the null and alternate hypothesis.

State the $p\text{-value}$.

State $\alpha$.

What is your decision?

Write a conclusion.

Answer any other questions asked in the problem.

According to the Center for Disease Control website, in 2011 at least 18% of high school students have smoked a cigarette. An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized–approximately 1,200 students–small city demographic) to determine if the local high school’s percentage was lower. One hundred fifty students were chosen at random and surveyed. Of the 150 students surveyed, 82 have smoked. Use a significance level of 0.05 and using appropriate statistical evidence, conduct a hypothesis test and state the conclusions.

A recent survey in the N.Y. Times Almanac indicated that 48.8% of families own stock. A broker wanted to determine if this survey could be valid. He surveyed a random sample of 250 families and found that 142 owned some type of stock. At the 0.05 significance level, can the survey be considered to be accurate?

$H_{0}: p = 0.488$ $H_{a}: p \neq 0.488$
$p\text{-value} = 0.0114$
$\alpha = 0.05$
Reject the null hypothesis.
At the 5% level of significance, there is enough evidence to conclude that 48.8% of families own stocks.
The survey does not appear to be accurate.

Driver error can be listed as the cause of approximately 54% of all fatal auto accidents, according to the American Automobile Association. Thirty randomly selected fatal accidents are examined, and it is determined that 14 were caused by driver error. Using $\alpha = 0.05$, is the AAA proportion accurate?

The US Department of Energy reported that 51.7% of homes were heated by natural gas. A random sample of 221 homes in Kentucky found that 115 were heated by natural gas. Does the evidence support the claim for Kentucky at the $\alpha = 0.05$ level in Kentucky? Are the results applicable across the country? Why?

$H_{0}: p = 0.517$ $H_{0}: p \neq 0.517$
$p\text{-value} = 0.9203$.
$\alpha = 0.05$.
Do not reject the null hypothesis.
At the 5% significance level, there is not enough evidence to conclude that the proportion of homes in Kentucky that are heated by natural gas is 0.517.
However, we cannot generalize this result to the entire nation. First, the sample’s population is only the state of Kentucky. Second, it is reasonable to assume that homes in the extreme north and south will have extreme high usage and low usage, respectively. We would need to expand our sample base to include these possibilities if we wanted to generalize this claim to the entire nation.

For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. The library director in Owensboro, Kentucky feels this is not true, so she asked a local college statistic class to conduct a survey. The class randomly selected 100 patrons and found that 82 borrowed books. Did the class demonstrate that the percentage was higher in Owensboro, KY? Use $\alpha = 0.01$ level of significance. What is the possible proportion of patrons that do borrow books from the Owensboro Library?

The Weather Underground reported that the mean amount of summer rainfall for the northeastern US is at least 11.52 inches. Ten cities in the northeast are randomly selected and the mean rainfall amount is calculated to be 7.42 inches with a standard deviation of 1.3 inches. At the $\alpha = 0.05 level$, can it be concluded that the mean rainfall was below the reported average? What if $\alpha = 0.01$? Assume the amount of summer rainfall follows a normal distribution.

$H_{0}: \mu \geq 11.52$ $H_{a}: \mu < 11.52$
$p\text{-value} = 0.000002$ which is almost 0.
At the 5% significance level, there is enough evidence to conclude that the mean amount of summer rain in the northeaster US is less than 11.52 inches, on average.
We would make the same conclusion if alpha was 1% because the $p\text{-value}$ is almost 0.

A survey in the N.Y. Times Almanac finds the mean commute time (one way) is 25.4 minutes for the 15 largest US cities. The Austin, TX chamber of commerce feels that Austin’s commute time is less and wants to publicize this fact. The mean for 25 randomly selected commuters is 22.1 minutes with a standard deviation of 5.3 minutes. At the $\alpha = 0.10$ level, is the Austin, TX commute significantly less than the mean commute time for the 15 largest US cities?

A report by the Gallup Poll found that a woman visits her doctor, on average, at most 5.8 times each year. A random sample of 20 women results in these yearly visit totals

3; 2; 1; 3; 7; 2; 9; 4; 6; 6; 8; 0; 5; 6; 4; 2; 1; 3; 4; 1

At the $\alpha = 0.05$ level can it be concluded that the sample mean is higher than 5.8 visits per year?

$H_{0}: \mu \leq 5.8$ $H_{a}: \mu > 5.8$
$p\text{-value} = 0.9987$
At the 5% level of significance, there is not enough evidence to conclude that a woman visits her doctor, on average, more than 5.8 times a year.

According to the N.Y. Times Almanac the mean family size in the U.S. is 3.18. A sample of a college math class resulted in the following family sizes:

5; 4; 5; 4; 4; 3; 6; 4; 3; 3; 5; 5; 6; 3; 3; 2; 7; 4; 5; 2; 2; 2; 3; 2

At $\alpha = 0.05$ level, is the class’ mean family size greater than the national average? Does the Almanac result remain valid? Why?

The student academic group on a college campus claims that freshman students study at least 2.5 hours per day, on average. One Introduction to Statistics class was skeptical. The class took a random sample of 30 freshman students and found a mean study time of 137 minutes with a standard deviation of 45 minutes. At α = 0.01 level, is the student academic group’s claim correct?

$H_{0}: \mu \geq 150$ $H_{0}: \mu < 150$
$p\text{-value} = 0.0622$
$\alpha = 0.01$
At the 1% significance level, there is not enough evidence to conclude that freshmen students study less than 2.5 hours per day, on average.
The student academic group’s claim appears to be correct.

9.7: Hypothesis Testing of a Single Mean and Single Proportion

Null and Alternative Hypothesis: Research Guidelines

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When undertaking a qualitative or quantitative research project, researchers must first formulate a research question, from which they develop their theories. By definition, an assumption is a prediction that a researcher makes about an actual research question and can either be affirmative or negative. In this case, writing a research question has three main components: variables (independent and dependent), a population sample, and the relation between these variables. To find null and alternative hypotheses, scholars identify a specific research question, determine the variables involved, and state H0 as no effect or difference and H1 or Ha as a significant effect or difference. When the prediction contradicts the research question, it is referred to as a null assumption. In short, an initial theory is a statement that implies there is no relationship between independent and dependent variables. Hence, researchers need to learn how to write a good null and alternative hypothesis to present quality studies.

General Aspects

Students with qualitative or quantitative research assignments must learn how to formulate and write good research questions and proposition statements. In essence, hypothesis testing is a statistical method used to determine if there is enough evidence to reject an initial theory and support an alternative assumption based on sample data. By definition, a research proposition is an assumption or prediction that a scholar makes before undertaking an experimental investigation. Basically, academic standards require such a prediction to be a precise and testable statement, meaning that researchers must prove or disapprove of it in the course of their assignment and provide alternatives if possible. In this case, the main components of a typical research assumption are variables (independent and dependent), a population sample, and the relation between these variables. To formulate a null hypothesis (H0) in quantitative research, researchers state there is no effect or difference between variables (e.g., µ1 = µ2), and, for an alternative hypothesis (H1 or Ha), they posit there is a significant effect or difference (e.g., µ1 ≠ µ2). Therefore, a research proposition is a prediction that scholars write about the relationship between two or more variables. In turn, a standard research inquiry is a particular process that seeks to answer a specific research question and, in the process, test a particular theory by confirming or disapproving it.

How to write a null and alternative hypothesis

Types of Hypotheses

There are several types of hypotheses, including null, alternative, directional, and non-directional assumptions. Basically, a directional hypothesis is a prediction of how an independent variable affects a dependent variable. In contrast, a non-directional hypothesis predicts that an independent variable influences a dependent variable but does not specify how. Regardless of the type, all propositions are about predicting the relationship between independent and dependent variables. To write H0 (null assumption) and H1 or Ha (alternative prediction), researchers clearly state H0 as a central assumption of no effect or no difference (e.g., µ1 = µ2) and H1 or Ha as a secondary assumption of a significant effect or difference (e.g., µ1 ≠ µ2).

What Is a Null Hypothesis (H0) and Its Purpose

According to its definition, a null hypothesis is a foundational statement in statistical testing that posits there is no significant effect, relationship, or difference between groups or variables within a given study. In simple words, a null hypothesis, usually symbolized as “H0,” is a statement that contradicts an actual research theory (Watt & Collins, 2019). The main purpose of writing a null hypothesis is to provide a basis for comparison, allowing researchers to determine whether there is sufficient evidence to reject this assumption in favor of an alternative theory, which suggests a real effect or relationship. As such, it is a negative statement, indicating that there is no relationship or connection between independent and dependent variables (Harrison et al., 2020). By starting with a null proposition, researchers can also employ various statistical tests to evaluate an entire data, ensuring the objectivity of findings and minimizing their bias. The process helps to ensure the validity of scientific research, minimizing the likelihood of drawing incorrect conclusions from the data collected. Moreover, by testing an initial theory, researchers can determine whether the inquiry results are due to the chance or the effect of manipulating a dependent variable (McNulty, 2022). In most instances, a null assumption corresponds with an alternative theory, a positive statement that covers a relationship that exists between independent and dependent variables. Finally, it is highly recommended that researchers should write an alternative assumption first before a null proposition.

What Is an Alternative Hypothesis (H1 or Ha) and Its Purpose

According to its definition and opposite to a null assumption, an alternative hypothesis in research is another statement in statistical testing that suggests there is a significant effect, relationship, or difference between groups or variables in a given study. Basically, this statement contrasts with what a null theory posits, which asserts that no such effect or relationship exists (Baker, 2021). The main purpose of writing an alternative hypothesis is to guide researchers in testing and validating new theories or effects and determine whether the observed data can provide evidence against a null proposition. The process involves comparing observed results to what would be expected under a null assumption. When statistical tests provide enough evidence to reject an initial postulation, an alternative theory becomes true, indicating that the observed effect or relationship is likely real and not due to random variation (Jawlik, 2016). By framing their research around an alternative hypothesis, scientists can focus their investigations on discovering meaningful effects and relationships, thereby advancing knowledge and understanding in their study fields. Hence, writing good null and alternative hypotheses is important because they provide a structured framework for statistical testing, allowing researchers to objectively evaluate evidence and draw conclusions about the presence of significant effects or relationships in an entire data.

Null vs. Alternative Hypothesis Formats

Steps on how to write a good null and alternative hypothesis.

Identify a Specific Research Question: Start with clearly defining a particular problem or phenomenon you want to study.
Determine Key Variables: Identify independent and dependent variables involved in your study.
State a Specific Null Hypothesis (H0): Formulate a concrete statement that suggests no effect, no difference, or no relationship between your variables. This is usually a statement of equality (e.g., µ1 = µ2).
State a Clear Alternative Hypothesis (H1 or Ha): Formulate another statement that suggests a significant effect, difference, or relationship between your variables. This is usually a statement of inequality (e.g., µ1 ≠ µ2, µ1 > µ2, or µ1 < µ2).
Means: H0: µ1 = µ2 vs. H1: µ1 ≠ µ2
Proportions: H0: p1 = p2 vs. H1: p1 ≠ p2
One-tailed test: If you are testing for a specific direction of effect (e.g., H1: µ1 > µ2).
Two-tailed test: If you are testing for any difference, regardless of direction (e.g., H1: µ1 ≠ µ2).
Consult Literature: Review existing research to see how similar or alternative theories have been formulated. This can provide guidance and ensure your expectations are aligned with standard practices in your field.
Write in Simple Terms: Ensure both null and alternative theories are stated clearly and concisely, making them easy to understand.
Review and Refine: Double-check your propositions for clarity and correctness. Make sure they are mutually exclusive and collectively exhaustive, covering all possible outcomes.
Seek Feedback: Discuss your approaches with peers or advisors to ensure they are logical, relevant, and testable. Adjust as necessary based on their input.

Note: A null hypothesis is a specific statement assuming no effect or difference, while other hypotheses refer to general statements that include writing null and alternative hypotheses and proposing possible outcomes to be tested.

Written Examples of Research Questions With H0 and H1 Hypotheses

Before developing any study proposition, a researcher must formulate a specific research question. In this case, a research hypothesis is a broad, testable statement about the expected relationship between variables, while a statistical hypothesis specifically refers to writing null and alternative hypotheses used in statistical testing to validate or refute an initial study assumption (O’Donnell et al., 2023). Then, the next step is to transform this study question into a negative statement that claims the lack of a relationship between independent and dependent variables. Alternatively, researchers can change the question into a positive statement that includes a relationship that exists between the variables. In turn, this latter statement becomes an alternative hypothesis and is symbolized as H1 or Ha. Hence, some of the examples of research questions and hull and alternative hypotheses are as follows:

Research Question (RQ) 1: Do physical exercises help individuals to age gracefully?

A Null Hypothesis (H0): Physical exercises are not a guarantee for graceful old age.
An Alternative Hypothesis (H1): Engaging in physical exercises enables individuals to remain healthy and active into old age.

RQ 2: What are the implications of therapeutic interventions in the fight against substance abuse?

H0: Therapeutic interventions are of no help in the fight against substance abuse.
H1: Exposing individuals with substance abuse disorders to therapeutic interventions helps to control and even stop their addictions.

RQ 3: How do sexual orientation and gender identity affect the experiences of late adolescents in foster care?

H0: Sexual orientation and gender identity have no effects on the experiences of late adolescents in foster care.
H1: The reality of stereotypes in society makes sexual orientation and gender identity factors complicate the experiences of late adolescents in foster care.

RQ 4: Does income inequality contribute to crime in high-density urban areas?

H0: There is no correlation between income inequality and incidences of crime in high-density urban areas.
H1: The high crime rates in high-density urban areas are due to the incidence of income inequality in those areas.

RQ 5: Does placement in foster care impact individuals’ mental health?

H0: There is no correlation between being in foster care and having mental health problems.
H1: Individuals placed in foster care experience anxiety and depression at one point in their life.

RQ 6: Do assistive devices and technologies lessen the mobility challenges of older adults with a stroke?

H0: Assistive devices and technologies do not provide any assistance to the mobility of older adults diagnosed with a stroke.
H1: Assistive devices and technologies enhance the mobility of older adults diagnosed with a stroke.

RQ 7: Does race identity undermine classroom participation?

H0: There is no correlation between racial identity and the ability to participate in classroom learning.
H1: Students from racial minorities are not as active as white students in classroom participation.

RQ 8: Do high school grades determine future success?

H0: There is no correlation between how one performs in high school and their success level in life.
H1: Attaining high grades in high school positions one for greater success in the future personal and professional lives.

RQ 9: Does critical thinking predict academic achievement?

H0: There is no correlation between critical thinking and academic achievement.
H1: Being a critical thinker is a pathway to academic success.

RQ 10: What benefits does group therapy provide to victims of domestic violence?

H0: Group therapy does not help victims of domestic violence because individuals prefer to hide rather than expose their shame.
H1: Group therapy provides domestic violence victims with a platform to share their hurt and connect with others with similar experiences.

Symbols and Signs in Writing

Common Mistakes

Ambiguity in Theories: Writing vague or unclear null and alternative assumptions.
Directional vs. Non-Directional Confusion: Confusing one-tailed (directional) and two-tailed (non-directional) claims.
Using Sample Statistics: Stating initial and alternative propositions in terms of sample statistics instead of population parameters.
Overlapping Assumptions: Creating null and alternative statements that are not mutually exclusive.
Testing Multiple Variables: Including multiple variables or conditions in a single theory.
Misinterpreting a Null Proposition: Assuming an initial statement is what you want to prove.
Incorrect Symbols and Signs: Using incorrect or inconsistent symbols and signs for writing null and alternative propositions.
Ignoring Context: Writing initial and alternative theories that are not relevant to an assigned research question or context.
Not Testable Hypotheses: Formulating null and alternative statements that are not testable with the available data or methods.
Confusing Null and Alternative Hypotheses: Swapping the roles of null and alternative assumptions.

The formulation of research questions in qualitative and quantitative assignments helps students to develop a specific theory for their experiments. In this case, learning how to write a good null and alternative hypothesis helps students and researchers to make their research relevant. Basically, the difference between a null and alternative hypothesis is that the former contradicts an entire research question, while the latter affirms it. In short, an initial proposition is a negative statement relative to a particular research question, and an alternative theory is a positive assumption. Moreover, it is important to note that developing a null hypothesis at the beginning of the assignment is for prediction purposes. As such, the research work must answer a specific research question and confirm or disapprove of an initial proposition. Hence, some of the tips that students and researchers need to know when developing any theory include:

Formulate a research question that specifies the relationship between an independent variable and a dependent variable.
Develop an alternative assumption that says a relationship exists between the variables.
Develop a null proposition that says a relationship does not exist between the variables.
Conduct an experiment to answer a research question under analysis, which allows the confirmation of a disapproval of a null theory or considering alternative options.

Baker, L. (2021). Hypothesis testing: How to choose the correct test (Getting started with statistics) . Chi-Squared Innovations.

Harrison, A. J., McErlain-Naylor, S. A., Bradshaw, E. J., Dai, B., Nunome, H., Hughes, G. T. G., Kong, P. W., Vanwanseele, B., Vilas-Boas, J. P., & Fong, D. T. (2020). Recommendations for statistical analysis involving null hypothesis significance testing. Sports Biomechanics , 19 (5), 561–568. https://doi.org/10.1080/14763141.2020.1782555

Jawlik, A. (2016). Statistics from A to Z: Confusing concepts clarified . John Wiley & Sons, Inc.

McNulty, R. (2022). A logical analysis of null hypothesis significance testing using popular terminology. BMC Medical Research Methodology , 22 (1), 1–9. https://doi.org/10.1186/s12874-022-01696-5

O’Donnell, C. T., Fielding-Singh, V., & Vanneman, M. W. (2023). The art of the null hypothesis — Considerations for study design and scientific reporting. Journal of Cardiothoracic and Vascular Anesthesia , 37 (6), 867–869. https://doi.org/10.1053/j.jvca.2023.02.026

Watt, R., & Collins, E. (2019). Null hypothesis testing . SAGE Publications Ltd.

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Two-Tailed Hypothesis Tests: 3 Example Problems

In statistics, we use hypothesis tests to determine whether some claim about a population parameter is true or not.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter = ≤, ≥ some value

H A (Alternative Hypothesis): Population parameter <, >, ≠ some value

There are two types of hypothesis tests:

One-tailed test : Alternative hypothesis contains either < or > sign
Two-tailed test : Alternative hypothesis contains the ≠ sign

In a two-tailed test , the alternative hypothesis always contains the not equal ( ≠ ) sign.

This indicates that we’re testing whether or not some effect exists, regardless of whether it’s a positive or negative effect.

Check out the following example problems to gain a better understanding of two-tailed tests.

Example 1: Factory Widgets

Suppose it’s assumed that the average weight of a certain widget produced at a factory is 20 grams. However, one engineer believes that a new method produces widgets that weigh less than 20 grams.

To test this, he can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

H 0 (Null Hypothesis): μ = 20 grams
H A (Alternative Hypothesis): μ ≠ 20 grams

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The engineer believes that the new method will influence widget weight, but doesn’t specify whether it will cause average weight to increase or decrease.

To test this, he uses the new method to produce 20 widgets and obtains the following information:

n = 20 widgets
x = 19.8 grams
s = 3.1 grams

Plugging these values into the One Sample t-test Calculator , we obtain the following results:

t-test statistic: -0.288525
two-tailed p-value: 0.776

Since the p-value is not less than .05, the engineer fails to reject the null hypothesis.

He does not have sufficient evidence to say that the true mean weight of widgets produced by the new method is different than 20 grams.

Example 2: Plant Growth

Suppose a standard fertilizer has been shown to cause a species of plants to grow by an average of 10 inches. However, one botanist believes a new fertilizer causes this species of plants to grow by an average amount different than 10 inches.

To test this, she can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

H 0 (Null Hypothesis): μ = 10 inches
H A (Alternative Hypothesis): μ ≠ 10 inches

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The botanist believes that the new fertilizer will influence plant growth, but doesn’t specify whether it will cause average growth to increase or decrease.

To test this claim, she applies the new fertilizer to a simple random sample of 15 plants and obtains the following information:

n = 15 plants
x = 11.4 inches
s = 2.5 inches
t-test statistic: 2.1689
two-tailed p-value: 0.0478

Since the p-value is less than .05, the botanist rejects the null hypothesis.

She has sufficient evidence to conclude that the new fertilizer causes an average growth that is different than 10 inches.

Example 3: Studying Method

A professor believes that a certain studying technique will influence the mean score that her students receive on a certain exam, but she’s unsure if it will increase or decrease the mean score, which is currently 82.

To test this, she lets each student use the studying technique for one month leading up to the exam and then administers the same exam to each of the students.

She then performs a hypothesis test using the following hypotheses:

H 0 : μ = 82
H A : μ ≠ 82

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The professor believes that the studying technique will influence the mean exam score, but doesn’t specify whether it will cause the mean score to increase or decrease.

To test this claim, the professor has 25 students use the new studying method and then take the exam. He collects the following data on the exam scores for this sample of students:

t-test statistic: 3.6586
two-tailed p-value: 0.0012

Since the p-value is less than .05, the professor rejects the null hypothesis.

She has sufficient evidence to conclude that the new studying method produces exam scores with an average score that is different than 82.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing What is a Directional Hypothesis? When Do You Reject the Null Hypothesis?

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Alternative Hypothesis: Definition, Types and Examples

In statistical hypothesis testing, the alternative hypothesis is an important proposition in the hypothesis test. The goal of the hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting the credibility of the alternative hypothesis instead of the default assumption made by the null hypothesis.

Null-Hypothesis-and-Alternative-Hypothesis

Both hypotheses include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research. In statistics, alternative hypothesis is often denoted as H a or H 1 .

Table of Content

What is a Hypothesis?

Alternative hypothesis, types of alternative hypothesis, difference between null and alternative hypothesis, formulating an alternative hypothesis, example of alternative hypothesis, application of alternative hypothesis.

"A hypothesis is a statement of a relationship between two or more variables." It is a working statement or theory that is based on insufficient evidence.

While experimenting, researchers often make a claim, that they can test. These claims are often based on the relationship between two or more variables. "What causes what?" and "Up to what extent?" are a few of the questions that a hypothesis focuses on answering. The hypothesis can be true or false, based on complete evidence.

While there are different hypotheses, we discuss only null and alternate hypotheses. The null hypothesis, denoted H o , is the default position where variables do not have a relation with each other. That means the null hypothesis is assumed true until evidence indicates otherwise. The alternative hypothesis, denoted H 1 , on the other hand, opposes the null hypothesis. It assumes a relation between the variables and serves as evidence to reject the null hypothesis.

Example of Hypothesis:

Mean age of all college students is 20.4 years. (simple hypothesis).

An Alternative Hypothesis is a claim or a complement to the null hypothesis. If the null hypothesis predicts a statement to be true, the Alternative Hypothesis predicts it to be false. Let's say the null hypothesis states there is no difference between height and shoe size then the alternative hypothesis will oppose the claim by stating that there is a relation.

We see that the null hypothesis assumes no relationship between the variables whereas an alternative hypothesis proposes a significant relation between variables. An alternative theory is the one tested by the researcher and if the researcher gathers enough data to support it, then the alternative hypothesis replaces the null hypothesis.

Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

There are a few types of alternative hypothesis that we will see:

1. One-tailed test H 1 : A one-tailed alternative hypothesis focuses on only one region of rejection of the sampling distribution. The region of rejection can be upper or lower.

Upper-tailed test H 1 : Population characteristic > Hypothesized value
Lower-tailed test H 1 : Population characteristic < Hypothesized value

2. Two-tailed test H 1 : A two-tailed alternative hypothesis is concerned with both regions of rejection of the sampling distribution.

3. Non-directional test H 1 : A non-directional alternative hypothesis is not concerned with either region of rejection; rather, it is only concerned that null hypothesis is not true.

4. Point test H 1 : Point alternative hypotheses occur when the hypothesis test is framed so that the population distribution under the alternative hypothesis is a fully defined distribution, with no unknown parameters; such hypotheses are usually of no practical interest but are fundamental to theoretical considerations of statistical inference and are the basis of the Neyman–Pearson lemma.

the differences between Null Hypothesis and Alternative Hypothesis is explained in the table below:

Formulating an alternative hypothesis means identifying the relationships, effects or condition being studied. Based on the data we conclude that there is a different inference from the null-hypothesis being considered.

Understand the null hypothesis.
Consider the alternate hypothesis
Choose the type of alternate hypothesis (one-tailed or two-tailed)

Alternative hypothesis must be true when the null hypothesis is false. When trying to identify the information need for alternate hypothesis statement, look for the following phrases:

"Is it reasonable to conclude..."
"Is there enough evidence to substantiate..."
"Does the evidence suggest..."
"Has there been a significant..."

When alternative hypotheses in mathematical terms, they always include an inequality ( usually ≠, but sometimes < or >) . When writing the alternate hypothesis, make sure it never includes an "=" symbol.

To help you write your hypotheses, you can use the template sentences below.

Does independent variable affect dependent variable?

Null Hypothesis (H 0 ): Independent variable does not affect dependent variable.
Alternative Hypothesis (H a ): Independent variable affects dependent variable.

Various examples of Alternative Hypothesis includes:

Two-Tailed Example

Research Question : Do home games affect a team's performance?
Null-Hypothesis: Home games do not affect a team's performance.
Alternative Hypothesis: Home games have an effect on team's performance.
Research Question: Does sleeping less lead to depression?
Null-Hypothesis: Sleeping less does not have an effect on depression.
Alternative Hypothesis : Sleeping less has an effect on depression.

One-Tailed Example

Research Question: Are candidates with experience likely to get a job?
Null-Hypothesis: Experience does not matter in getting a job.
Alternative Hypothesis: Candidates with work experience are more likely to receive an interview.
Alternative Hypothesis : Teams with home advantage are more likely to win a match.

Some applications of Alternative Hypothesis includes:

Rejecting Null-Hypothesis : A researcher performs additional research to find flaws in the null hypothesis. Following the research, which uses the alternative hypothesis as a guide, they may decide whether they have enough evidence to reject the null hypothesis.
Guideline for Research : An alternative and null hypothesis include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research.
New Theories : Alternative hypotheses can provide the opportunity to discover new theories that a researcher can use to disprove an existing theory that may not have been backed up by evidence.

We defined the relationship that exist between null-hypothesis and alternative hypothesis. While the null hypothesis is always a default assumption about our test data, the alternative hypothesis puts in all the effort to make sure the null hypothesis is disproved.

Null-hypothesis always explores new relationships between the independent variables to find potential outcomes from our test data. We should note that for every null hypothesis, one or more alternate hypotheses can be developed.

Also Check:

Mathematics Maths Formulas Branches of Mathematics

FAQs on Alternative Hypothesis

What is hypothesis.

A hypothesis is a statement of a relationship between two or more variables." It is a working statement or theory that is based on insufficient evidence.

What is an Alternative Hypothesis?

Alternative hypothesis, denoted by H 1 , opposes the null-hypothesis. It assumes a relation between the variables and serves as an evidence to reject the null-hypothesis.

What is the Difference between Null-Hypothesis and Alternative Hypothesis?

Null hypothesis is the default claim that assumes no relationship between variables while alternative hypothesis is the opposite claim which considers statistical significance between the variables.

What is Alternative and Experimental Hypothesis?

Null hypothesis (H 0 ) states there is no effect or difference, while the alternative hypothesis (H 1 or H a ) asserts the presence of an effect, difference, or relationship between variables. In hypothesis testing, we seek evidence to either reject the null hypothesis in favor of the alternative hypothesis or fail to do so.

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Alternative Hypothesis: Powerful Insights, Challenges and 5 Essential Steps to Develop One

Introduction.

In the realm of research, the alternative hypothesis, a key type of research hypothesis , plays a pivotal role in shaping the direction and outcomes of scientific inquiry. Understanding the alternative hypothesis is essential for researchers, as it not only guides the formulation of research questions but also influences the choice of statistical tests and the interpretation of results. This article delves into the significance of the alternative hypothesis in research, offering insights and examples that illustrate its critical function in the research process.

What is the Alternative Hypothesis in Research?

The alternative hypothesis, often denoted as Ha or H1, plays a crucial role in the realm of research methodology. It serves as a counterpoint to the null hypothesis , which posits that there is no effect or relationship between the variables being studied. In contrast, the alternative hypothesis asserts that there is indeed an effect or a relationship, suggesting that the observed data deviates from what would be expected under the null hypothesis. This fundamental distinction is essential for researchers as it guides the direction of their inquiry and the interpretation of their findings.

The alternative hypothesis proposes that there is a significant effect or relationship between variables, guiding researchers to explore and validate their assumptions in a study.

In statistical hypothesis testing, the alternative hypothesis is not merely a statement; it embodies the researcher’s expectations based on prior knowledge or theoretical frameworks . It is a declaration that there is a significant difference or correlation between the variables in question. This hypothesis is critical because it sets the stage for statistical testing, where researchers seek to gather evidence to support or refute it, ultimately contributing to the body of knowledge in their field.

The Significance of the Alternative Hypothesis in Research

Guiding research with a well-defined alternative hypothesis.

The alternative hypothesis plays a crucial role in research methodology as it serves as the foundation for statistical testing. By proposing a specific effect or relationship between variables, it allows researchers to explore and validate their assumptions. This hypothesis is essential for guiding the research design, as it helps in determining the appropriate data collection methods and analysis techniques.

Clear Alternative Hypothesis for Accurate Research Findings

Without a well-defined alternative hypothesis, researchers may struggle to interpret their findings or may inadvertently overlook significant relationships that could contribute to their field of study. A clear hypothesis supports meaningful analysis, ensuring that important relationships and findings are accurately identified and understood.

Encouraging Critical Thinking

The alternative hypothesis fosters a deeper understanding of the phenomena being investigated. It encourages researchers to think critically about their predictions and the implications of their results, which in turn sharpens their focus and clarifies the relevance of their study outcomes.

Assessing Evidence Through Hypothesis Comparison

By contrasting the alternative hypothesis with the null hypothesis, which asserts that no effect exists, researchers can effectively assess the strength of their evidence. This comparison enhances the rigor of the research, helping to clarify whether observed effects are due to real relationships or mere chance.

Catalyzing Inquiry and Innovation in Research

In essence, the alternative hypothesis is not just a statement; it is a catalyst for inquiry and innovation. It challenges existing theories, promotes critical analysis, and opens new avenues for discoveries, ultimately contributing to the advancement of knowledge in the field.

Application of Alternative Hypothesis in Quantitative Research

Example 1 : exploring the relationship between spending and tipping in restaurants.

To illustrate the concept of Ha in a quantitative research context, consider a study examining the relationship between the amount spent at a restaurant and the tip left by customers. The null hypothesis (H0) might state that there is no correlation between the bill amount and the tip percentage. In contrast, the Ha asserts that there is a statistically significant correlation, suggesting that as the bill amount increases, the tip percentage also rises. This assumption is grounded in the expectation that higher spending may lead to higher tipping, reflecting a common social norm.

Example 2 : Evaluating the Impact of New Medication on Blood Pressure

In another example, imagine a study investigating the effectiveness of a new medication on reducing blood pressure. Here, the null hypothesis would claim that the medication has no effect on blood pressure compared to a placebo. On the other hand, Ha proposes that the medication does lead to a significant reduction. This Ha is essential as it directs the research design, data collection, and statistical analysis, ultimately helping researchers assess the treatment’s effectiveness. By clearly defining Ha, researchers can focus on gathering evidence to either support or refute this expectation, thereby advancing scientific knowledge.

5 Steps To Develop a Strong Alternative Hypothesis

Step 1 : conduct a thorough literature review.

Crafting a strong Ha begins with a comprehensive literature review . This step helps researchers identify existing gaps in knowledge, providing a foundation for formulating relevant research questions and understanding the context of the problem being investigated.

An alternative hypothesis emerges from thorough literature reviews, as researchers identify gaps in existing knowledge and propose a testable relationship or effect to explore.

Step 2 : Establish a Specific Research Question

Once the knowledge gaps are clear, researchers can establish a specific research question . This question serves as the basis for the alternative hypothesis, ensuring that it is directly tied to the study’s focus and purpose.

Formulating a research question using “what”, “where”, “when”, “why”, “who”, and “how” helps shape a focused alternative hypothesis by clarifying the specific relationships or effects to be tested.

Step 3 : Formulate a Clear and Concise Alternative Hypothesis

The alternative hypothesis should then be articulated as a clear statement reflecting the study’s expected outcome. Often framed in an if-then format, this structure helps enhance clarity and directs the research towards measurable outcomes.

Step 4 : Define Variables and Ensure Testability

A strong alternative hypothesis is both specific and testable. Researchers should define all variables involved, which will guide data collection methods and analysis , ensuring that the hypothesis can be empirically evaluated.

Step 5 : Differentiate Clearly from the Null Hypothesis

Lastly, the alternative hypothesis must be mutually exclusive from the null hypothesis—if one is true, the other must be false. This clear distinction allows for meaningful conclusions, strengthening the research’s contribution to the broader field.

By following these steps, researchers can create an effective alternative hypothesis that provides direction, ensures rigor, and adds value to their academic discipline.

Challenges in Defining the Alternative Hypothesis

Balancing specificity and testability.

One of the primary challenges in defining an alternative hypothesis is achieving a balance between specificity and testability. Researchers must ensure that the hypothesis is precise enough to offer clear direction for the study, yet broad enough to allow for various potential outcomes. This balance is crucial to avoid ambiguity and prevent overlap with the null hypothesis, which could make it challenging to draw definitive conclusions .

Avoiding Bias in Hypothesis Formulation

Another challenge in defining the alternative hypothesis is the potential for researcher bias . Unconscious expectations or reliance on previous findings may lead researchers to favor a particular outcome, unintentionally shaping the hypothesis to fit these expectations. Such bias can affect the objectivity of data interpretation, as results may be skewed to support the initial alternative hypothesis rather than being assessed on their own merit.

Addressing the Complexity of Real-World Phenomena

Real-world phenomena are often complex, and this can lead to the possibility of multiple alternative hypotheses for a single research question. Defining a single, clear alternative hypothesis may be difficult in such cases, as researchers need to carefully choose the hypothesis that best aligns with their research objectives . This complexity can complicate research design, analysis, and the interpretation of findings, adding another layer of difficulty to hypothesis development.

These challenges highlight the need for careful consideration and methodological rigor in defining an alternative hypothesis to ensure clarity, accuracy, and relevance in research findings.

In conclusion, the alternative hypothesis is fundamental in research methodology, acting as a counterpoint to the null hypothesis and guiding both the design and interpretation of studies. By clarifying what researchers aim to confirm or refute, the alternative hypothesis provides a foundation for statistical testing and analysis, helping to structure the research in a meaningful way. A well-defined hypothesis not only adds rigor to the study but also enhances the clarity and focus of the investigation.

However, developing a strong alternative hypothesis presents challenges, including potential bias and the risk of misinterpreting data to fit expected outcomes. Despite these challenges, a carefully constructed alternative hypothesis strengthens the validity of findings, encourages critical thinking, and deepens understanding of the research topic . Acknowledging both its strengths and limitations remains crucial for researchers aiming to contribute valuable insights and advance knowledge across various fields.

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Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

Null hypothesis (H 0 ): There’s no effect in the population .
Alternative hypothesis (H A ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

They’re both answers to the research question
They both make claims about the population
They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
Alternative hypothesis (H A ): Independent variable affects dependent variable .

Test-specific

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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