Handout 5

Handout 5 (Practice Final) 1. Go over the following Lecture Notes: • Introduction to Linear Regression (more weight) • Normal Distribution • Foundations for Inference • Inference for Categorical Data (more weight) • Inference for Numerical Data (more weight) 2. Review Quiz 2 thoroughly, answer Key posted on Canvas 3. Review Handouts 3-4 thoroughly, answer Key posted on Canvas 4. Review Homework’s 4-8 thoroughly, answer Key posted on Canvas 5. Suppose you are interested in measuring the amount of time on average it takes you to make your commute to school. Over 18 random days, you estimated that the average time is 38.4 minutes with a standard deviation of 5.362 minutes. Construct a 90% confidence interval for the mean commute time to school.

6. For a random sample sample of size 12, one has calculated the 95% confidence interval for µ and obtained the result (46.2, 56.9). a) What is the margin of error for this confidence interval? b) What is the point estimate for that sample? c) Find the standard deviation (s) for that sample.

8. A forester measured 20 of the trees in a large woods that is up for sale. He found that their mean diameter was 191 inches and their standard deviation 22.4 inches. Suppose that these trees provide an accurate description of the whole forest and that the diameter of the tree follows a normal distribution. Find the following: a) What percentage of trees would be above 191 inches in diameter? b) What percentage of trees would be between 180 and 190 inches? c) What size diameter would you say represents the top 20% of the trees?

9. Do teachers find their work rewarding and satisfying? An article reports the results of a survey of 395 elementary school teachers and 266 high school teachers. Of the elementary school teachers, 224 said they were very satisfied with their jobs, whereas 126 of the high school teachers were very satisfied with their work. Use alpha=5%. a) Estimate the difference between the proportion of all elementary school teachers who are very satisfied and all high school teachers who are very satisfied by calculating and interpreting a confidence interval (CI). b) Perform an appropriate statistical test to this problem and compare your answer to part(a).

11. Marijuana and Road Safety Gallup polled 635 Democrats and 372 Republicans on their opinion of driving while impaired by marijuana. 20% of Democrats and 35% of Republicans said that driving while impaired by marijuana poses a very serious threat, compared to alcohol, prescription painkillers, and prescription antidepressants. (a) Suppose we want to conduct a hypothesis test for evaluating whether the proportions of Democrats and Republicans who think that driving while impaired by marijuana poses a very serious threat are different. i. Calculate the sample statistic for this hypothesis test. ii. Calculate the standard error for this hypothesis test. iii. Perform the test and state your conclusion, use α = 0.05. (b) Suppose we want to estimate the difference between the proportions of Democrats and Republicans who think that driving while impaired by marijuana poses a very serious threat using a confidence interval. No calculation is required for the following questions. i. Is the value of the sample statistic for this confidence interval different than the sample statistic for the hypothesis test in part (a)? If yes, explain how and why. ii. Is the value of the standard error for this confidence interval different than the standard error for the hypothesis test in part (a)? If yes, explain how and why. iii. Construct and interpret a 95% confidence interval.

12. Using the Handout 1 dataset posted on Canvas. At 5% level of significance, perform appro- priate statistical analysis to test to see if there is a significant mean difference in commitment scores with respect to sex. Answer this question by clearly stating your hypotheses, test statis- tic, and conclusion. Use R to solve this problem.

16. Veterinarians at a nonhuman primate research center are interested in estimating the true average birth weight of rhesus monkeys born in captivity. Below are the summary statistics of the data and output from the analysis testing if the true average birth weight of the monkeys is 0.4kg. What is the correct calculation to estimate the true average birth weight of rhesus monkeys with a 95% confidence interval? min Q1 median Q3 max mean sd n missing 0.27 0.37 0.39 0.5 0.68 0.44 0.12 10 0 t = 1.0853, df = 9, p-value = 0.306 alternative hypothesis: true mean is not equal to 0.4 95 percent confidence interval: XXXXXXX XXXXXXX (a) 0 . 44 ± 1 . 0853 × 0 . 12 / √ 9 (b) 0 . 44 ± 1 . 0853 × 0 . 12 (c) 0 . 44 ± 2 . 26 × 0 . 12 / √ 10 (d) 0 . 39 ± 1 . 96 × 0 . 12 (e) 0 . 39 ± 2 . 26 × 0 . 12 / √ 9 17. For the following R output, which of the following is true ? t = 0.2024, df = 14, p-value = 0.8425 alternative hypothesis: true mean is not equal to 20 95 percent confidence interval: 18.91483 21.31132 sample estimates: mean of x 20.11307 (a) This is a one-sided test. (b) At α = 0 . 05, we reject the null hypothesis. (c) There is a 0.16 probability that the null hypothesis is false. (d) This analysis had a sample size of n = 14. (e) None of the above are true. 18. Suppose we conduct a one-sample test with H 0 : µ = 50 and H a : µ ̸ = 50. Given that ¯ x = 60, t = 3 . 0, and n = 21, what can we say about the p -value for the test? (a) 0 . 005 < p -value < 0 . 010 (b) p -value < 0 . 010 (c) p -value > 0 . 005 (d) 0 . 050 < p -value < 0 . 100 (e) 0 . 100 < p -value < 0 . 200

19. Based on a random sample of 120 rhesus monkeys, a 95% confidence interval for the proportion of rhesus monkeys that live in a captive breeding facility and were assigned to research studies is (0.67, 0.83). Which of the following is true ? (a) 95 of the sampled monkeys were assigned to research studies (b) the margin of error for the confidence interval is 0.16 (c) a larger sample size would yield a wider confidence interval (d) if we used a different confidence level, the interval would not be symmetric about the sample proportion (e) none of the above are true 20. Does acupuncture cure morning sickness? Researchers randomly randomly assigned 100 preg- nant women into two equal-sized groups: treatment and control. Patients in the treatment group received acupuncture that is specifically designed to treat morning sickness. Patients in the control group received placebo acupuncture (needle insertion at non-acupoint locations). Following the treatments patients were asked if their morning sickness symptoms improved. The proportion of patients who said they experienced improvement was 3% higher in the treat- ment group. To test whether this difference could be attributed to chance, a statistics student decided to conduct a randomization test. She represented each patient with an index card, and noted on the cards whether the patient experienced improvement or not. She shuffled the cards together very well, and then dealt them into two equal-sized groups. Which of the following best describes the outcome? (a) The difference in proportions of improvement between the two stacks of cards is expected to be 0%. (b) If acupuncture is effective, the difference in proportions of improvement between the two stacks of cards will be more than 3%. (c) If acupuncture is not effective, the difference in proportions of improvement between the two stacks of cards is expected to be 3%. (d) The difference in proportions of improvement between the two stacks of cards is expected to be 3%. 21. A test was performed to determine if the proportion of public water sources that are safe for consumption differs from 0.80. That is, we tested H 0 : p = 0 . 80 vs H a : p ̸ = 0 . 80. In 64 out of 70 sampled sites, the public water sources were safe for consumption, and the test yielded a p -value of 0.02. What conclusion can we draw regarding the proportion of public water sources that are safe for consumption at the α = 0 . 05 level of significance? (a) Because the p -value < α , we conclude that the proportion of public water sources that are safe for consumption does not significantly differ than 0.80. (b) Because the p -value > α , we conclude that the proportion of public water sources that are safe for consumption does not significantly differ than 0.80. (c) Because the p -value < α , we conclude that the proportion of public water sources that are safe for consumption is significantly different than 0.80. (d) Because the p -value > α , we conclude that the proportion of public water sources that are safe for consumption is significantly different 0.80. (e) Because the p -value < α , we conclude that the proportion of public water sources that are safe for consumption is significantly greater than 0.80.