Exam 2

Exam 2 1. (3 points) What does it mean when results of a certain test were “statistically significant”? a. It meant results as extreme as the observed sample statistic are unlikely to occur by random chance alone (assuming the null hypothesis is true) b. It meant results as extreme as the observed sample statistic are likely to occur by random chance alone (assuming the null hypothesis is true) c. It meant results are extreme as the observed sample statistic are unlikely to occur by random chance alone (assuming the alternative hypothesis is true) d. It meant results as extreme as the observed sample statistic are likely to occur by random chance alone (assuming the alternative hypothesis is true) 2. (3 points) Which are the correct null and alternative hypothesis for the following situation? Testing to see if there is evidence that the correlation between two variables is positive. a. ࠵? ! : ࠵? = 0 ࠵?࠵? ࠵? " : ࠵? ≠ 0 b. ࠵? ! > 0 ࠵?࠵? ࠵? " : ࠵? = 0 c. ࠵? ! = 0 ࠵?࠵? ࠵? " : ࠵? > 0 d. ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # > ࠵? $ 3. (3 points) Which are the correct null and alternative hypothesis for the following situation? Testing to see if average sales are different in stores where customers are approached by salespeople than in stores where they are not. Let group 1 be the group of stores where customers are approached by salespeople and group 2 be the group of stores where customers were not approached. a. ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # ≠ ࠵? $ b. ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # ≠ ࠵? $ c. ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # > ࠵? $ d. ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # > ࠵? $ 4. (3 points) Which of the following is a correct statistical hypothesis test? a. ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # < ࠵? $ b. ࠵? ! : ࠵?̅ = 7 ࠵?࠵? ࠵? " : ࠵?̅ ≠ 7 c. ࠵? ! : ࠵? # < ࠵? $ ࠵?࠵? ࠵? " : ࠵? # > ࠵? $ d. ࠵? ! : ࠵? > 0.5 ࠵?࠵? ࠵? " : ࠵? = 0.5 5. (3 points) What is a p-value? a. The chance that ࠵? ! is true. b. The chance that ࠵? " is true. c. The chance of the test saying ࠵? " is true when really ࠵? ! is true. d. Chance of obtaining a sample statistic as extreme (or more extreme) than the observed sample statistic, if the null hypothesis is true. 6. (3 points) Of the p-values below, which one provides the strongest evidence against ࠵? ! ? a. ࠵? − ࠵?࠵?࠵?࠵?࠵? = 0.17 b. ࠵? − ࠵?࠵?࠵?࠵?࠵? = 0.03 c. ࠵? − ࠵?࠵?࠵?࠵?࠵? = 0.28 d. ࠵? − ࠵?࠵?࠵?࠵?࠵? = 0.51

7. (6 points) The following randomization distribution is for the test ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # > ࠵? $ . Suppose ࠵?̅ # − ࠵?̅ $ = 8 . What is the corresponding p-value and the correct generic conclusion about ࠵? ! using ࠵? = 0.01 ? p-value: Generic conclusion (circle your answer): Reject ࠵? ! Fail to reject ࠵? ! Accept ࠵? ! 8. (3 points) What is the conclusion of a test based on a p-value of 0.123 with a 10% significance level? a. Do not reject ࠵? ! b. Reject ࠵? ! c. We need more information to decide d. Accept ࠵? ! 9. (3 points) What is a Type II Error? a. Not rejecting a false null b. Not rejecting a true null c. Rejecting a false null d. Rejecting a true null 10. (3 points) Where is a randomization distribution centered? a. The value of the sample statistic b. It depends on the sample c. The hypothesized value of the parameter in the null hypothesis d. You do not know where a randomization distribution will be centered 11. (3 points) Reallocating cases to treatment groups while keeping the response values the same, making Ho true then bootstrapping, and randomly permuting/scrambling/shuffling one variable are all ways to create what? a. Creating confidence intervals b. Making randomization distributions

Read the following excerpt from an online article from Newswise and answer questions 16-20. Legislators struggle to distinguish between Al and constituents. 22-Mar-2023 11:35 AM EDT, by Cornell University Newswise — ITHACA, N.Y. - Natural language models, such as ChatGPT and GPT-4, open new opportunities for malicious actors to influence representative democracy, new Cornell University research suggests. A field experiment investigating how the natural language model GPT-3, the predecessor to the most recently released model, might be used to generate constituent email messages showed that legislators were only slightly less likely to respond to Al-generated messages (15.4%) than human-generated (17.3%). The 2% difference, gleaned from more than 32,000 messages sent to about 7,000 state legislators in the U.S., was statistically significant but substantially small, the researchers said. 16. (6 points) Define the parameters and state the null and alternative hypothesis for the test comparing the response rates between AI-generated messages and human-generated messages. ࠵? ! : ࠵? " : 17. (4 points) Give the notation and value for the sample statistic. 18. (3 points) Which outcome was chosen, and how do you know? a. ࠵? ! because the article is not statistically significant b. ࠵? ! because the article is statistically significant c. ࠵? " because the article is not statistically significant d. ࠵? " because the article is statistically significant 19. (3 points) What do we know about the p-value compared to alpha? a. ࠵? − ࠵?࠵?࠵?࠵?࠵? ≥ ࠵? because they failed to reject ࠵? ! b. ࠵? − ࠵?࠵?࠵?࠵?࠵? ≥ ࠵? because they rejected ࠵? ! c. ࠵? − ࠵?࠵?࠵?࠵?࠵? < ࠵? because they failed to reject ࠵? ! d. ࠵? − ࠵?࠵?࠵?࠵?࠵? < ࠵? because they rejected ࠵? !

20. (3 points) Which error are we possibly making here? a. Type I b. Type II c. Both d. We can’t tell from this example. 21. (3 points) The J.D. Power 2021 U.S. Vehicle Dependability Study™ (VDS) reported the average number of problems per 100 vehicles for many different brands. The reported number of problems for Chevrolet was 115 per 100 vehicles and the reported number for Mercedes-Benz was 122 problems per 100 vehicles (the industry average was 121). On average, what is the difference between Chevrolet and Mercedes-Benz per vehicle? Is it practically significant? a. The difference per car is 0.07 problems, which is not practically significant. b. The difference per car is 0.07 problems, which is practically significant. c. The difference per car is 7 problems, which is not practically significant. d. The difference per car is 7 problems, which is practically significant. 22. (3 points) Suppose the normal curve shown to the right is approximately a randomization distribution for a test of the form ࠵? ! : ࠵? # = ࠵? $ ࠵?࠵? ࠵? " : ࠵? # ≠ ࠵? $ . Which of the following values of ࠵?̅ # − ࠵?̅ $ would provide the least amount of evidence against ࠵? ! and in favor of ࠵? " ? a. A b. B c. C d. D 23. (3 points) The z*value for a 90% confidence interval is 1.645. Which of the options below is most likely the z*value for an 85% confidence interval? a. 1.645 b. 1.960 c. 2.330 d. 1.440

Use the following StatKey output to answer the next two questions. (5 points) Compute a 95% confidence interval for the true average ࠵? (z* is 1.96 for 95% confidence). Show your work and give the lower and upper bounds. Round your answer to two decimal places. (6 points) Compute the standardized test statistic for the test ࠵? ! : ࠵? = 31,000 ࠵?࠵? ࠵? " : ࠵? ≠ 31,000 . Show your work and round your answer to two decimal places. Using the interval you found in the previous problem, will you reject or fail to reject this test if there is significance level ࠵? = 0.05 ? Page 2 3 4 5 6 7 8 Total Score Total 18 12 18 13 15 13 11 100