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?
Come up with a real-world example where you would:
1. Reject a true null hypothesis
2. Fail to reject a false null hypothesis
Explain the difference between the null hypothesis ($ H_0 $) and the alternative hypothesis ($ H_A $). Which one do we test first?

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

How does choosing your significance level $ \alpha $ relate to the risk of a Type I error?
Why does decreasing the probability of making a type one error increase the probability of making a type two error?
How does a researcher decide the level of significance for a hypothesis test?
The mean price of mid-sized cars in a region is $32,000. A test is conducted to see if the claim is true. State the Type I and Type II errors in complete sentences.
A sleeping bag is tested to withstand temperatures of –15 °F. You think the bag cannot stand temperatures that low. State the Type I and Type II errors in complete sentences.
A machine in a factory is supposed to fill vials with approximately 50mL 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 the Null and Alternative Hypotheses to the scenario using the correct symbols
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?
State the Type I and Type II errors in complete sentences given the following statements.
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 San Jose State University graduates is at least $100,000 per year.
4. Twenty-nine percent of high school seniors get drunk each month.
5. Fewer than 5% of adults ride the bus to work in Los Angeles.
6. The mean number of cars a person owns in his or her 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 of six weeks.
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.
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 decide whether to reject or fail to reject the null hypothesis?
Why do we make our decision about $ \alpha $ before seeing the results of the test?
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?
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. State the null and alternative hypotheses.
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 50mL 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. Suppose that we fail to reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made.
2. Suppose that we reject the null hypothesis. Write a conclusion statement in the context of the problem. Assume no errors were made
Why can we never accept the null hypothesis?
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. Write an appropriate conclusion.
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.
Write the null and alternative hypotheses in wrds and then symbols for each of 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. 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?
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. \[H₀ : \bar {x} < 2 hours\]
2. \[H_A : \bar {x} > 3.5 hours\]

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

Which two distributions can you use for hypothesis testing for this chapter?
Which distribution do you use when you are testing a population mean and the standard deviation is known? Assume sample size is large.
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? Assume the underlying population is normal.
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. State the hypotheses
2. Calculate the z statistic
3. Find the p-value
4. Make a decision at $ \alpha = 0.05 $
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 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 , 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 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?
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 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.
How do you tell whether the test is left, right, or two tailed?
Explain in your own words why knowing the population standard deviation is important for using a z-test.
True or False: “A large positive z-score always means we reject the null.” Explain.
What does an extremely small p-value (e.g., 0.001) suggest about our result?
What is a practical situation where rejecting $ H_0 $ would lead to a big decision (e.g., policy change, funding reshuffle)?
A hospital administrator mentioned 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 $\alpha = 0.01$?
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? We have verified the independence, sample size, and skew conditions are satisfied.

8.4: Comparing Two Means (Independent Samples)

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 (use tech or calculator)
4. Conclusion?
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.
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.
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.
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?

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 42% of respondents in a taste test would prefer Brand A. In a particular test of 100 people, 39% preferred Brand A. What distribution should you use to perform a hypothesis test?
You are performing a hypothesis test of a single population proportion. The data come from which distribution?
You are performing a hypothesis test of a single population mean using a Student’s -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. Test the claim at $ \alpha = 0.05 $.
1. State hypotheses
2. Find the sample proportion
3. Compute test statistic
4. Decision and interpretation
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.
"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.

8.6: Interpreting p-values and Significance

Explain in your own words what a p-value means. What does it assume?
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.
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.
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? The -value is 0.0264. State the null and alternative hypotheses and interpret the p-value.
What should you do if \alpha\ equals the p-value?

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

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