"Math Sample Final Exam Questions Fall On the math"
Math 57 Sample Final Exam Questions Fall 2005 1. On the math SAT men have a distinct edge. In 1994, for example, the men averaged about 500, and the women averaged about 460. The SD for both groups was about 100. (a) Estimate the percentage of men getting over 600 on this test in 1994. (b) Estimate the percentage of women getting over 600 on this test in 1994. (c) What did you assume about the SAT scores in order to compute the above values? 2. A survey is carried out by the ﬁnance department to determine the distribution of household size in a certain city. They draw a simple random sample of 1,000 households. After several visits, the interviewers ﬁnd people at home in only 653 households. Rather than face such a high non-response rate, the department draws a new batch of households, and uses the ﬁrst 347 completed interviews in the second batch together with the 653 in order to have a sample of 1,000 households. The department counts 3,087 people in these 1,000 households, and estimates the average household size in the city to be about 3.1 persons. Is this estimate likely to be too low, too high, or about right? Explain. 3. A study on college students found that the men had an average weight of about 66kg and a SD of 9kg. The women had an average weight of about 55kg and a SD of 9kg. (a) Find the averages and SDs in pounds (assume that 1kg = 2lbs ... even though it is closer to 1kg=2.2lbs) (b) Just roughly, what percentage of men weigh between 57kg and 75kg? (c) If you took the men and women together, would the SD of their weights be smaller than 9kg, bigger than 9kg, or just about 9kg? Explain. 4. The diameter of the moon is measured four times independently by a process that is free of bias. The individual measurements came out 2157, 2166, 2162, 2155 miles, which average out to 2160. One more measurement is about to be taken using the same process. When compared with the estimate of 2160 miles, you would expect this new measurement to be: (Explain) (a) more accurate as a measure of the true diameter of the moon? (b) less accurate as a measure of the true diameter of the moon? (c) about as accurate as a measure of the true diameter of the moon? 5. For 24 diﬀerent brands of cigarettes, the nicotine content (in mg) and tar content (in mg) was measured. The regression line is: nicotine content (mg) = 0.132 + 0.062 * tar content (mg) (a) Interpret the values 0.132 and 0.062. What are their units? What do they say about the rela- tionship between nicotine content and tar content? (b) For a cigarette with 10mg of tar, how many mg of nicotine should you expect in the cigarette? 6. Suppose you have to cross a train track on your commute. The probability that you will have to wait for a train is 0.2. If you don’t have to wait, the commute takes 15 minutes, but if you have to wait, it takes 20 minutes. (a) What is the expected value for the time it takes you to commute? (b) Is the expected time ever the actual commute time? Explain. 1 7. One large course has 900 students, broken down into section meetings with 30 students each. The section meetings are led by TAs. On the ﬁnal, the class average is 63 and the SD is 20. However, in one section the average is only 55. The TA argues this way: If you took 30 students at random from the entire class, there is a pretty good chance that they would average below 55 on the ﬁnal. That’s what happened to me – chance variation. Is this a good defense? Explain. (Hint: you don’t really need to do a hypothesis test here, but a p-value and the interpretation that goes along with the p-value will help anwer the question.) 8. One hundred investigators set out to test a diﬀerent null hypothesis. Unknown to them, all the null hypotheses happen to be true. Investigator #1 gets a p-value of 0.55. Investigator #2 gets a p-value of 0.08. And so forth. How many investigators should get a statistically signiﬁcant result? Explain 9. For each of the following, specify the “parameter” and “speciﬁc value” we are interested in testing: (a) Do a majority of Americans between the ages of 18 and 30 think the use of marijuana should be legalized? (b) Is the mean of the Math SAT scores in California in a given year diﬀerent from the target mean of 500 set by the test developers? 10. A tire manufacturer believes that the treadlife of its tires can be described by a normal model with a mean of 32,000 miles and a standard deviation of 2,500 miles. (a) You believe that, because of the way you drive, you will get 34,000 miles out of your tires. What percent of people get more treadlife out of their tires than you do? (b) Find the IQR (inner-quartile range) of the treadlifes. Show your work. 11. We would like to know: “What is the average price of a house in Franklin County, OH?” We took a random sample of 100 houses, and found the average to be $120,128 and the standard deviation to be $50,324. Use the data to provide an answer to the question. 12. Bill, a psychologist, has developed a new aptitude test. He was able to show (through a somewhat dubious study) that 80% of the public passed the test. A second psychologist, Tamera, wants to recreate this study. She believes the actual percentage is less than 80%. A random sample of 200 people took the test, and 152 passed. does Tamera have enough evidence to say that Bill’s test is invalid? Justify your answer with a complete hypothesis test. 13. Data were collected in a statistics class of 35 students on how many countries and states were visited by each member of the class. The instructor wants to use this data to predict how many countries a student has visited based on the number of states he or she has visited, and obtains the following regression population model and output. # countries visited = βo + β1 · # states visited coef st.dev.(coef) Constant 0.592 1.747 # States visited 0.241 0.076 (a) Do # of countries visited and # of states visited have a linear relationship? Test the appropriate hypothesis. (b) Consider the population of all students who have visited the exact same number of states as you have. If you were to predict (1) the number of countries those people have visited on average or (2) the number of countries a randomly selected individual has visited, which would be more variable? That is, if you make prediction intervals for both (1) and (2) which interval would be wider? Explain. 2