STAT 220 Midterm Exam, Friday, Feb. 24 Name Please show all of your work on the exam itself. If you need more space, use the back of the page. Remember that partial credit will be awarded when appropriate. This is a closed-book exam, and no calculators are allowed. All calculations can be done by hand, but if you get stuck on calculations, then at least you should write your answer in unsimplified form. Problem 1. [2 out of 20 points] Consider the sample 1, 1, 1, 5. (a) Give the first three numbers of the five-number summary. Explain briefly how you obtain these three numbers. (b) Find the sample standard deviation. Show your work. (Hint: Calculating this number does not require a calculator.)
Problem 2. [2 out of 20 points] Each of the following R commands contains a mistake. Give a brief explanation of the mistake, then write a correct version that fixes the mixtake. (a) # Obtain FALSE TRUE FALSE FALSE sqrt(c(4, 16, 25, 36)) = 4 (b) # Create a vector of names fruit = c(apple, berry, cherry)
Problem 3. [3 out of 20 points] Suppose that women s heights are normally distributed with a mean of 65 inches and a standard deviation of 2.5 inches. (a) Find the probability that a randomly selected woman will be between 60 and 67.5 inches tall. Show all work. (b) If a simple random sample of three women is selected, what is the probability that all three of them will be shorter than 65 inches? Show all work. (c) Suppose a simple random sample of size n = 25 women is selected. Describe, as completely as you can, the sampling distribution of the sample mean height, X.
Problem 4. [3 out of 20 points] It is believed that regular physical exercise leads to a lower resting pulse rate. Following are summary data for n = 25 individuals on resting pulse rate and whether the individuals regularly exercises or not. Assuming this is a random sample from a larger population, use this sample to determine whether the mean pulse is lower for those who exercise. Use the unpooled variance formula, and clearly show all steps of the hypothesis test. (Hint: All the calculations here can be done by hand.) The steps you should show are these: (a) writing appropriate hypotheses; (b) calculating the test statistic; (c) finding the p-value; and (d) writing your conclusion using plain language, for which you may use the 0.05 level. Exercise? n mean SD Yes 9 62.0 12.0 No 16 77.0 12.0
Problem 5. [2 out of 20 points] For each of the following 4 situations, write S, N, or R (for selection, nonresponse, or response ) to describe the most serious form of bias that may result. Give a brief, one-sentence explanation for each answer. (a) A list of registered automobile owners is used to select a random sample for a survey about whether people think homeowners should pay a surtax to support public parks. (b) In a college town, college students are hired to conduct door-to-door interviews, based on a multistage cluster sample, to determine whether city residents think there should be a law forbidding loud music at parties. (c) The 1936 Literary Digest survey (d) A random sample of incoming Penn State freshmen is selected to be interviewed, together with their parents while visiting campus for FTCAP orientation, about first-year students attitudes about underage drinking at Penn State. Problem 6. [1 out of 20 points] Answer either (i) OR (ii) but not both (you decide). (i) When testing H 0 : p 1 = p 2, the standard error of (ˆp 1 ˆp 2 ) is given by ˆp(1 ˆp)( 1 n 1 + 1 n 2 ). Give a formula for ˆp. (There is more than one way to write this formula; you may use any form you choose.) (ii) Give a formula for s p, the pooled standard deviation in a situation involving the difference of two means from independent samples.
Problem 7. [4 out of 20 points] A sample of college students was asked if they would return the money if they found a wallet on the street. Of the 100 women, 90 said yes, and of the 100 men, 80 said yes. Assume these students represent all college students. (a) Find a 95% confidence interval for the proportion of college men who would say yes to this question. (Hint: All the calculations here can be done by hand.) (b) Find an approximate 95% confidence interval for the difference in the proportions of college men and women who would say yes to this question. (Hint: All the calculations here can be done by hand.)
Problem 8. [2 out of 20 points] Suppose that Alicia takes a test to determine whether she has a particular disease. We know that this test is 99% accurate for people who do have the disease, and the test is 91% accurate for people who do not have the disease. (In other words, 99% of people with the disease will test positive, and 9% of people without the disease will test positive.) Suppose we also know that only 1% of individuals truly have this disease. If Alicia tests positive, what is the probability that she has the disease? (Hint: No calculator should be required to express the correct probability as a percentage.) [IF you are unable to answer the question above, you can try this question instead for only 1 point: What proportion of individuals will test positive for this disease?] Problem 9. [1 out of 20 points] How many different strings of 12 coin flips in a row will result in a total of 3 heads? Show all your work, and express your final answer as a single integer.