Final Exam Spring 2023

Name & UNI: Stats 1101, Spring 2023 Final Exam This exam will not com pletely re flect the 2023 Fall semester course be cause of the change in text book and course struc ture. Vo cab u lary like “lurk ing vari able” and con cepts like QQ - plots will not be tested. z -table z +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 χ 2 distribution critical values α 0.010 0.025 0.05 0.10 d.f. 1 6.6349 5.0239 3.8415 2.7055 2 9.2103 7.3778 5.9915 4.6052 3 11.3449 9.3484 7.8147 6.2514 4 13.2767 11.1433 9.4877 7.7794 5 15.0863 12.8325 11.0705 9.2364 6 16.8119 14.4494 12.5916 10.6446 t distribution critical values (two-tailed) α 0.01 0.05 0.10 d.f. 10 3.1693 2.2281 1.8125 20 2.8453 2.0860 1.7247 30 2.7500 2.0423 1.6973 40 2.7045 2.0211 1.6839 50 2.6778 2.0086 1.6759

Following the instructions on the previous page counts for three points. Multiple Choice Mark your answer in your blue book. Binary choice questions are each worth 2 points. The other questions are worth 5 points each. Figure 1: (above) 1. Figure 1 shows a symmetric distribution. A. True B. False 2. Based on Figure 1 , the mean will be greater than the median. A. True B. False 3. In a randomized controlled experiment, the researcher knows and controls how the assignment of factor levels (or treatment and control) is done. A. True B. False 4. A response variable is any variable positively correlated with an explanatory variable. A. True B. False 5. A researcher compares the rate of cycling accidents among children who wore a bi- cycle helmet at some point in a 24-hour period to children who never wore a helmet in that period. The researcher reports that wearing a helmet is associated with a 10% increase in bicycle accidents. A plausible lurking variable is A. the baseline rate of bicycles accidents. B. bicycle use. Page 2

B. s ≥ 1 14. What is the sample variance for the values -2, 0, 2? A. 2 B. 4 15. We conduct a two-sided z test and find a test statistic of .04. For α = .05, do we reject the null hypothesis? A. Yes, reject the null. B. No, fail to reject the null. 16. We conduct a two-sided z test and find a P-value of 0.04. For α = .05, do we reject the null hypothesis? A. Yes, reject the null. B. No, fail to reject the null. 17. Hospital records show that 15% of all patients are admitted for surgical treatment, 8% are admitted for obstetrics, and 2% receive both obstetrics and surgical treatment. What percentage are admitted for surgery or obstetrics? A. 21% B. 25% 18. Hospital records show that 15% of all patients are admitted for surgical treatment, 8% are admitted for obstetrics, and 2% receive both obstetrics and surgical treatment. Are these events (admission for surgical treatment and admission for obstetrics) in- dependent? A. Yes, independent. B. No, not independent. 19. Which of these is not an assumption for simple linear regression? A. r xy ̸ = 0 B. There is a linear relationship between x and y . 20. Which of these is not an assumption for inference for linear regression? A. r xy > 0.5 B. The residuals are approximately normally distributed. 21. Suppose that ˆ y = 1 + x 1 but that in alternative models, ˆ y = 1 + 0 x 1 + 2 x 2 and that ˆ y = 1 + 2 x 2 . A. The residuals from ˆ x 1 = b 0 + b 1 x 2 have no linear relationship with the resid- uals from ˆ y = 1 + 2 x 2 . Page 4

B. The residuals from ˆ x 2 = b 0 + b 1 x 1 have no linear relationship with the resid- uals from ˆ y = 1 + x 1 . 22. QQ-plots are only valid when the data is normally distributed. A. True B. False 23. After conducting a one-sided hypothesis test with a null hypothesis of µ = 2, you find a P-value of 0.99. Suppose the sample mean was 1. Which of these could have been the alternative hypothesis? A. µ > 2 B. µ < 2 24. Which of these is most indicative of an overfit model? A. The correlation coefficient between x and y is r = 1. B. The R 2 falls substantially when calculating it using test data vs when using training data. Use the following information for the next five questions. X is drawn from a normal distribution with mean zero and standard deviation one. Y is equal to X if X is positive. If X is negative, then Y = 1 − X . 25. X and Y are independent. A. True B. False 26. What is P ( X > 1)? A. 0.16 B. 0.84 27. What is P ( Y > 1 2 | | X | > 1 2 )? A. 1 2 B. 1 28. What is P ( X < 2) A. .0250 B. .0500 C. .9500 D. .9772 E. .9902 29. What is P ( Y > 2 | | X | > 0.675)? Page 5

C. The dependent variable only takes on two values. D. The independent variable only takes on two values. E. All of the above. Use the following information for the next four questions. An experiment is run where some participants receive therapy, some receive a puppy, some receive both therapy and a puppy, and a control group receives nothing. Re- searchers are interested in the effect on y , a measure of antisocial behavior, where a higher y means more antisocial behavior. Consider three models fit to the data. Each controls for participant income, because one treatment group may have a higher av- erage income. ˆ y = a 0 + a 1 Income + a 2 TherapyOnly + a 3 PuppyOnly + a 4 TherapyAndPuppy (1) ˆ y = b 0 + b 1 Income + b 2 Therapy + b 3 Puppy (2) ˆ y = c 0 + c 1 Income + c 2 TherapyAndOrPuppy (3) 34. Using model 1, what is the predicted value of y for a participant with Income = 1 who was in the therapy-only group? A. a 0 B. a 0 + a 2 C. a 0 + a 1 + a 2 D. a 1 + a 2 E. a 2 35. Which of these is necessarily true? A. a 1 = b 1 B. a 4 = a 2 + a 3 C. b 2 + b 3 = c 2 D. b 2 = a 2 − a 3 E. None of the above Open Response 36. (5 points) Refer again to the setup for question 34. Suppose a 4 < 0. What is the interpretation? You do not need to address statistical significance. 37. (5 points) Refer again to the setup for question 34. Make an argument for why the R 2 from model 1 should be at least as high as the R 2 from model 3. Page 7

38. (5 points) Randomized experiments are generally considered to be more trustworthy than observational studies as causal evidence. Give one reason with a brief explana- tion. 39. (4 points) Describe a practical reason why researchers might rely on observational data instead of an experiment. 40. (5 points) Jonathan argues that social media is bad for mental health. Noah argues that social media isn’t inherently dangerous, but that having it accessible on a smart- phone is a problem. An experiment had some participants deactivate their social media accounts and others continued with normal social media use. The assignment of social media vs. no social media was done randomly. At the end, those who deac- tivated their accounts were shown to have better mental health. Does the experiment provide strong evidence for either’s claim? Provide a brief explanation. 41. (20 points) Students can fail a course because of deficient understanding or because of difficult circumstances. A student of deficient understanding will fail 100% of the time. Students who understand the material will encounter bad luck that causes them to fail 10% of the time. Think of this as a hypothesis test. Let the null hypothesis be that a student understands the material. When a student fails, we can say that we’ve rejected the null hypothesis. a.) A student of deficient understanding will fail 100% of the time. What statistical concept does this correspond to? You do not have to provide an explanation. b.) A student who understands the material will fail 10% of the time. What statisti- cal concept does this correspond to? You do not have to provide an explanation. c.) A professor concludes that, because a student failed, a student can be judged as having deficient understanding with 100% certainty. What kind of mistake is this? Provide a brief explanation that describes the mistake. d.) Calculate the probability that a failing student has a deficient understanding if we start with a belief that 10% of students will have a deficient understanding. Show your work. 42. (30 points) Researchers want to know if a preference for dogs or cats is related to the preference for coffee or tea. Researchers ask people leaving a convention center in Las Vegas. The researcher’s population of interest is all residents of Las Vegas. Sixteen out of thirty-two respondents prefer coffee to tea. Eighteen out of thirty-two respondents prefer cats to dogs. a.) Construct two different 2 × 2 contingency tables that are consistent with the above. b.) Assuming independence between the two variables, find the expected cell counts. c.) Explain if each of the assumptions for a χ 2 test are met, including if the sample is representative of the population of interest. Page 8