8.8: Practice (Chapter 8)

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8.1: Why Hypothesis Testing and Type I and Type II Errors

What type of statistical question is best suited for a hypothesis test, and why?
For each of the following, determine if the statement is a Null Hypothesis. If it is not, rewrite it so that it becomes a Null Hypothesis.
1. The mean number of years Americans work before retiring is 34.
2. At most 60% of Americans vote in presidential elections.
3. The mean starting salary for Red Rocks Community College graduates is not equal to that of Community College of Denver graduates.
4. Twenty-nine percent of high school seniors get drunk each month.
5. Fewer than 5% of adults ride the bus to work in Denver.
6. The mean number of cars a person owns in their lifetime is not more than ten.
7. About half of Americans prefer to live away from cities, given the choice.
8. Europeans have a mean paid vacation each year equal to that of Asians.
9. The chance of developing breast cancer is under 11% for women.
10. Private universities mean tuition cost is more than $20,000 per year.
11. The mean price of mid-sized cars in a region is $32,000.
12. A sleeping bag is tested to withstand temperatures of –15 °F.
Write the null and hypothesis in words for the following situations.
1. New York is known as "the city that never sleeps". A random sample of 25 New Yorkers were asked how much sleep they get per night. Do these data provide convincing evidence that New Yorkers on average sleep less than 8 hours a night?
2. Employers at a company are worried about the effect of March Madness, a basketball championship held each spring in the US, on employee productivity. They estimate that on a regular business day employees spend on average 15 minutes of company time checking personal email, making personal phone calls, etc. They also collect data on how much company time employees spend on such non-business activities during March Madness. They want to determine if these data provide convincing evidence that employee productivity decreases during March Madness.
3. The mean entry level salary of an employee at a company is $58,000. You believe it is higher for IT professionals in the company.
Write the null and hypothesis in words for the following situation.
1. Since 2008, chain restaurants in California have been required to display calorie counts of each menu item. Prior to menus displaying calorie counts, the average calorie intake of diners at a restaurant was 1100 calories. After calorie counts started to be displayed on menus, a nutritionist collected data on the number of calories consumed at this restaurant from a random sample of diners. Do these data provide convincing evidence of a difference in the average calorie intake of a diners at this restaurant?
2. Based on the performance of those who took the GRE exam between July 1, 2004 and June 30, 2007, the average Verbal Reasoning score was calculated to be 462. In 2011 the average verbal score was slightly higher. Do these data provide convincing evidence that the average GRE Verbal Reasoning score has changed since 2004?

Fill out the following error table. Label which boxes represent Type I error, Type II error, and correct decisions:

Hypothesis Testing Outcomes
Reality (Truth)	We Reject $ H_0 $	We Fail to Reject $ H_0 $
$ H_0 $ is true
$ H_0 $ is false

Food inspectors inspect samples of food products to see if they are safe. This can be thought of as a hypothesis test with the null hypothesis "The food is safe". The following is an example of what type of error? The sample suggests that the food is safe, but it actually is not safe.
You conduct a hypothesis test to determine whether most people believe in UFOs. Given a null hypothesis of: Most people believe in UFOs, the result of your test is to fail to reject. In fact, most people do not believe in UFOs. In this case, you have made what what type of error?
Come up with a real-world example where you would:
1. Reject a true null hypothesis
2. Fail to reject a false null hypothesis
What is a practical situation where rejecting $ H_0 $ would lead to a big decision (e.g., policy change, funding reshuffle)?
Why does decreasing the probability of making a type one error increase the probability of making a type two error?
A machine in a factory is supposed to fill vials with approximately 50 mL of a liquid. The quality control manager wants to test whether the machine is working properly. If the machine is found to be over or under-filling then the assembly line will need to be shut down so that it can be re-calibrated.
1. Write the Null and Alternative Hypotheses to the scenario in words.
2. What would it mean to make a Type 1 error in this situation?
3. What would it mean to make a Type 2 error in this situation?
4. Which error might be worse in this scenario and why?
A patient named Diana was diagnosed with Fibromyalgia, a long-term syndrome of body pain, and was prescribed anti-depressants. Being the skeptic that she is, Diana didn't initially believe that anti-depressants would help her symptoms. However after a couple months of being on the medication she decides that the anti-depressants are working, because she feels like her symptoms are in fact getting better.
1. Write the hypotheses in words for Diana's skeptical position when she started taking the anti-depressants.
2. What is a Type 1 error in this context?
3. What is a Type 2 error in this context?
4. How would these errors affect the patient?
A food safety inspector is called upon to investigate a restaurant with a few customer reports of poor sanitation practices. The food safety inspector uses a hypothesis testing framework to evaluate whether regulations are not being met. If he decides the restaurant is in gross violation, its license to serve food will be revoked.
1. Write the hypotheses in words.
2. What is a Type 1 error in this context?
3. What is a Type 2 error in this context?
4. Which error is more problematic for the restaurant owner? Why?
5. Which error is more problematic for the diners? Why?
6. As a diner, would you prefer that the food safety inspector requires strong evidence or very strong evidence of health concerns before revoking a restaurant's license? Explain your reasoning.
Suppose regulators monitored 403 drugs last year, each for a particular adverse response. For each drug they conducted a single hypothesis test with a significance level of 5% to determine if the adverse effect was higher in those taking the drug than those who did not take the drug; the regulators ultimately rejected the null hypothesis for 42 drugs.
1. Describe the error the regulators might have made for a drug where the null hypothesis was rejected.
2. Describe the error regulators might have made for a drug where the null hypothesis was not rejected.
3. Suppose the vast majority of the 403 drugs do not have adverse effects. Then, if you picked one of the 42 suspect drugs at random, about how sure would you be that the drug really has an adverse effect?
4. Can you also say how sure you are that a particular drug from the 361 where the null hypothesis was not rejected does not have the corresponding adverse response?

8.2: Hypothesis Testing Framework

List the five steps in the hypothesis testing process, and write one sentence describing the purpose of each.
What factors determine which test statistic (z, t, etc.) you should use?
How do you tell whether a test is left, right, or two tailed?
How does the type of test e.g., left tailed, relate to the inequality symbols used in the hypotheses?
How does a researcher decide the level of significance for a hypothesis test?
Why do we make our decision about $ \alpha $ before seeing the results of the test?
How do you decide whether to reject or fail to reject the null hypothesis?
Why can we never accept the null hypothesis?
Complete this prompt: “When we reject the null hypothesis, we are concluding that...”
What are some cases where “failing to reject” $ H_0 $ might still give you useful information?
Write the null and alternative hypotheses using equations/inequalities with the correct symbols for the following situations.
1. New York is known as "the city that never sleeps". A random sample of 25 New Yorkers were asked how much sleep they get per night. Do these data provide convincing evidence that New Yorkers on average sleep less than 8 hours a night?
2. Employers at a company are worried about the effect of March Madness, a basketball championship held each spring in the US, on employee productivity. They estimate that on a regular business day employees spend on average 15 minutes of company time checking personal email, making personal phone calls, etc. They also collect data on how much company time employees spend on such non-business activities during March Madness. They want to determine if these data provide convincing evidence that employee productivity decreases during March Madness.
3. The mean entry level salary of an employee at a company is $58,000. You believe it is higher for IT professionals in the company.
Write the null and alternative hypotheses using equations/inequalities with the correct symbols for the following situations.
1. Since 2008, chain restaurants in California have been required to display calorie counts of each menu item. Prior to menus displaying calorie counts, the average calorie intake of diners at a restaurant was 1100 calories. After calorie counts started to be displayed on menus, a nutritionist collected data on the number of calories consumed at this restaurant from a random sample of diners. Do these data provide convincing evidence of a difference in the average calorie intake of a diners at this restaurant?
2. Based on the performance of those who took the GRE exam between July 1, 2004 and June 30, 2007, the average Verbal Reasoning score was calculated to be 462. In 2011 the average verbal score was slightly higher. Do these data provide convincing evidence that the average GRE Verbal Reasoning score has changed since 2004?
3. A statistics instructor believes that fewer than 20% of Red Rocks Community College students attended the opening night midnight showing of the latest Harry Potter movie. They survey 84 of their students and finds that 11 of them attended the midnight showing.
A study suggests that the average college student spends 2 hours per week communicating with others online. You believe that this is an underestimate and decide to collect your own sample for a hypothesis test. You randomly sample 60 students from your dorm and nd that on average they spent 3.5 hours a week communicating with others online. A friend of yours, who offers to help you with the hypothesis test, comes up with the following set of hypotheses. Indicate any errors you see.
1. $H_0 : \bar{x} < 2$ hours
2. $H_A : \bar{x} > 3.5$ hours
The proportion of Americans who have frequent migraines is 15.2% according to the CDC. An acupuncturist claims that her treatment can reduce this figure significantly. A random sample of 279 Americans is administered the acupuncturists treatment and 39 report experiencing migraines.
1. State hypotheses for the scenario using the correct symbols.
2. Suppose that we fail to reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made.
3. Suppose that we reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made
A machine in a factory is supposed to fill vials with approximately 50 mL of a liquid. The quality control manager wants to test whether the machine is working properly. If the machine is found to be over or under-filling then the assembly line will need to be shut down so that it can be re-calibrated.
1. State hypotheses for the scenario using the correct symbols.
2. Suppose that we fail to reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made.
3. Suppose that we reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made
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. Write an appropriate conclusion.
1. State hypotheses for the scenario using the correct symbols.
2. Suppose that we fail to reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made.
3. Suppose that we reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made

8.3: Tests for a Single Mean (z and t-tests)

Which two distributions can you use for a hypothesis test for a single mean?
Which distribution do you use when you are testing a population mean and the standard deviation is known? Assume sample size is large.
Explain in your own words why knowing the population standard deviation is important for using a z-test.
You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.
A population mean is 13. The sample mean is 12.8, and the sample standard deviation is two. The sample size is 20. What distribution should you use to perform a hypothesis test? What assumptions should be made?
It is believed that Red Rocks Community College (RRCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 RRCC 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 RRCC Intermediate Algebra students get less than seven hours of sleep per night, on average? What is the distribution to be used for this test?
A researcher claims that the average time people spend on social media each day is more than 2 hours. A sample of 50 people has an average of 2.3 hours and a known population standard deviation of 0.8 hours.
1. What type of test should the researcher conduct?
2. What significance level should the researcher choose?
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?
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 1% 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¢. Does the data support the claim at the 10% 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 the $ x= $ number of sick days they took for the past year. Should the personnel team believe that the mean number is ten?
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.
True or False: “A large positive z-score always means we reject the null.” Explain.
A hospital administrator randomly selected 64 patients and measured the time (in minutes) between when they checked in to the ER and the time they were first seen by a doctor. The average time is 137.5 minutes and the standard deviation is 39 minutes. He is getting grief from his supervisor on the basis that the wait times in the ER increased greatly from last year's average of 127 minutes. However, the administrator claims that the increase is probably just due to chance.
1. Are conditions for inference met? Note any assumptions you must make to proceed.
2. Using a significance level of $\alpha = 0.05$, is the change in wait times statistically significant? Use a two-sided test since it seems the supervisor had to inspect the data before he suggested an increase occurred.
3. Would the conclusion of the hypothesis test change if the significance level was changed to $\alpha = 0.01$?
Researchers investigating characteristics of gifted children collected data from schools in a large city on a random sample of thirty-six children who were identified as gifted children soon after they reached the age of four. The distribution of the ages (in months) at which these children first counted to 10 successfully is provided as sample statistics $ \bar{x}=30, \sigma=1 $
1. Are conditions for inference satisfied?
2. Suppose you read on a parenting website that children first count to 10 successfully when they are 32 months old, on average. Perform a hypothesis test to evaluate if these data provide convincing evidence that the average age at which gifted children first count to 10 successfully is different than the general average of 32 months. Use a significance level of 0.10.
3. Interpret the p-value in context of the hypothesis test and the data.
4. Calculate a 90% confidence interval for the average age at which gifted children first count to 10 successfully.
5. Do your results from the hypothesis test and the confidence interval agree? Explain.
A manufacturer claims that bearings produced by their machine last 7 hours on average under harsh conditions. A factory worker randomly samples 75 ball bearings, and records their lifespans under harsh conditions. They calculate a sample mean of 6.85 hours, and the standard deviation of the data is 1.25 working hours. Conduct a formal hypothesis test of this claim. Make sure to check that relevant conditions are satisfied.
The nutrition label on a bag of potato chips says that a one ounce (28 gram) serving of potato chips has 130 calories and contains ten grams of fat, with three grams of saturated fat. A random sample of 35 bags yielded a sample mean of 134 calories with a standard deviation of 17 calories. Is there evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips? Assume we have verified the independence, sample size, and skew conditions are satisfied.

8.4: Comparing Two Means (Independent Samples)

What does Welch’s t-test account for that a pooled two-sample t-test does not?
Do hypothesis tests for two means require equal sample sizes? Why or why not?
Give an example of a study that compares two means but would NOT be appropriate for a t-test. Explain why.
A fitness company is testing two diets. Group A lost an average of 5.1 lbs ($ s = 2.3 $, $ n = 15 $). Group B lost 4.2 lbs ($ s = 2.1 $, $ n = 15 $). Is there evidence that Diet A works better at a 5% significance level?
1. State the hypotheses
2. Compute the test statistic (use Welch’s t-test)
3. Find the approximate p-value using technology.
4. Write the conclusion in complete sentences.
A sample of 12 in-state graduate school programs at school A has a mean tuition of $64,000 with a standard deviation of $8,000. At school B, a sample of 16 in-state graduate programs has a mean of $80,000 with a standard deviation of $6,000. On average, is the mean tuition different?
A student at a four-year college claims that mean enrollment at four–year colleges is higher than at two–year colleges in the United States. Two surveys are conducted. Of the 35 two–year colleges surveyed, the mean enrollment was 5,068 with a standard deviation of 4,777. Of the 35 four-year colleges surveyed, the mean enrollment was 5,466 with a standard deviation of 8,191. On average, is the mean enrollment of four-year colleges higher than two-year colleges?

An economist wants to compare mean hourly rate charged by automobile mechanics in two suburbs. She randomly selects auto repair facilities from both suburbs and records their hourly rates (in dollars). The data are as follows:

Suburb 1 Data
35.3	33.2	37.8	35.6	33.5
36	36.1	31.7	29.6	37.6
37.9

Suburb 2 Data
47.7	50.7	42.2	41.9	48.7
42.9	42.4	37.3	38.2	45.3
44	43.8	42.4	44.1	42.7
48.6	56.2	46.3	51.6	38.4
51.1	47.5

Use 10% level of significance to decide whether there is sufficient evidence that the mean hourly rate charged by automobile mechanics in suburb 1 is the same as the mean hourly rate charged by automobile mechanics in suburb 2. You may also assume that the samples are from normal populations.

8.5: Tests for a Single Proportion

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses.
It is thought that more than 50% of respondents in a taste test would prefer Brand A. In a particular test of 100 people, 49 preferred Brand A. State the null and alternative hypotheses.
You are performing a hypothesis test of a single population mean using a Student’s t-distribution. What must you assume about the distribution of the data?
A policy maker claims that fewer than half of the city’s residents support a new tax. A random sample finds that 91 out of 200 residents support the tax.
1. Find the sample proportion
2. Compute test statistic
Explain each part of the z-test for a proportion: what are $ \hat{p} $, $ p_0 $, and $ n $?
Why do we use $ p_0 $ (the null value) in the denominator instead of $ \hat{p} $?
You are performing a hypothesis test of a single population proportion. What must be true about the quantities of \np\ and \nq\?
What are some situations in which you might prefer a two-tailed hypothesis about a proportion instead of a one-tailed one?
True or False: “If $ p = 0.04 $, that means 4% of the population is in favor.” Explain.
It is believed that nearsightedness affects about 8% of all children. In a random sample of 194 children, 21 are nearsighted.
1. Construct hypotheses appropriate for the following question: do these data provide evidence that the 8% value is inaccurate?
2. What proportion of children in this sample are nearsighted?
3. Given that the standard error of the sample proportion is 0.0195 and the point estimate follows a nearly normal distribution, calculate the test statistic (the Z statistic).
4. What is the p-value for this hypothesis test?
5. What is the conclusion of the hypothesis test?
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.
Your statistics instructor claims that 60 percent of the students who take her Intro to 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 Intro to Statistics students and find that 34 feel more enriched as a result of her class. Conduct a hypothesis test for the instructor's claim.
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? Conduct a hypothesis test for Newsweek's claim.
"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.
1. 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.
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.
Is the proportion of cat lovers in Lakewood more than the proportion in Golden? Out of 498 people surveyed in Lakewood, 284 are cat lovers. Out of 480 people surveyed in Golden, 263 are cat lovers. Conduct a full hypothesis test at the 0.05 level of significance to answer the question.

8.6: Interpreting p-values and Significance

Is the p-value for a Z-statistics of 2.5 larger when n=500 or n=1000?
Explain in your own words what a p-value means. What does it assume?
What does an extremely small p-value (e.g., 0.001) suggest about our result?
What is “p-hacking”? Why is it problematic in scientific research?
Give an example where a result could be statistically significant but not practically significant.
Draw a sketch of the sampling distribution and p-value for the following hypothesis test: $ H_0: \mu=23, H_1: \mu >23, \bar{x}=25$
Draw a sketch of the sampling distribution and p-value for the following hypothesis test: $ H_0: p=0.3, H_1: p \ne 0.3, \hat{p}=.25$
How does choosing your significance level $ \alpha $ relate to the risk of a Type I error?
What happens if you lower $ \alpha $ (say, from 0.05 to 0.01)? How does that affect:
1. Type I error risk?
2. Likelihood of rejecting $ H_0 $?
Explain how reporting only p-values (and not effect sizes, variability, or context) can be misleading.
Interpret: p = 0.075. Which is more correct?
1. There is fairly strong evidence against $ H_0 $
2. We should reject $ H_0 $
3. We fail to reject $ H_0 $ because we lack strong evidence
4. Explain your choice.
The mean age of graduate students at a University is at most 31 years with a standard deviation of two years. A random sample of 15 graduate students is taken. The sample mean is 32 years and the sample standard deviation is three years. Are the data significant at the 1% level?
1. The p-value is 0.0264. Interpret the p-value in complete sentences.
A USA Today/Gallup poll conducted between 2010 and 2011 asked a group of unemployed and underemployed Americans if they have had major problems in their relationships with their spouse or another close family member as a result of not having a job (if unemployed) or not having a full-time job (if underemployed). 27% of the 1,145 unemployed respondents and 25% of the 675 underemployed respondents said they had major problems in relationships as a result of their employment status.
1. The p-value for this hypothesis test is approximately 0.35. Explain what this means in context of the hypothesis test and the data.
A running shoe manufacturer believes that the proportion of faulty running shoes from the supplier is greater than 0.1, the proportion stated by the supplier. We perform a hypothesis test at a significance level of 0.05 with null and alternative hypotheses: H₀: p = 0.1 and H_a: p > 0.1. A researcher inspects 100 items and finds 16 faulty ones. She gives a p-value of 0.02 in her analysis. What is the correct interpretation of the p-value?
What should you do if \alpha\ equals the p-value?
You are given the following hypotheses: $ H_0: \mu = 34, H_A: \mu > 34 $. We know that the sample standard deviation is 10 and the sample size is 65. For what sample mean would the p-value be equal to 0.05? Assume that all conditions necessary for inference are satisfied.

Practice problems include parts of:

Introductory Statistics 2e by OpenStax is licensed under CC BY 4.0

Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

and

OpenIntro Statistics by David Diez, Christopher Barr, & Mine Çetinkaya-Rundel is licensed under CC BY 3.0

Reality (Truth)	We Reject \( H_0 \)	We Fail to Reject \( H_0 \)
\( H_0 \) is true
\( H_0 \) is false

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