AP Statistics Lectures
by Arnold Kling

Cumulative Practice Questions, week of December 17

Turn in questions (1)-(7) on Tuesday, questions (8)-(14) on Wednesday, and questions (15)-(20) on Thursday.

1. A system has four independent components--A, B, C, and D. The probability that each component works is given by P(A) = .95, P(B) = .90, P(C) = .99, and P(D) = .90. What is the probability that the entire system works properly (no component fails)? What is the probability that at least one of the four components works properly?

2. If P(A) = .4, P(B) =.2, and P(A and B) = .08, are A and B independent? Are they mutually exclusive?

3. What is the probability that a shipment of 100 fruit will have no more than 6 rotten fruits if the probability that any one fruit is rotten is .04?

4. If the probability that a hitter gets a base hit on any at bat is .27, what is the probability that the first base hit will occur on the fourth at bat? What is the expected number of at bats it will take to get the first hit?

5. Suppose that when you are tested for strep throat, the test is positive for 98 percent of people who have the disease. However, the test also is positive for 1 percent of people who do not have the disease. Suppose that in a large group of third-graders, 7 percent actually have strep throat.

What is the probability that a third-grader chosen at random will test positively for strep?

What is the probability that a third-grader who tests positively for strep actually will have the disease?

Comment on whether you think this is an effective test.

6. It has been observed over many samples that the probability that more than 3 eggs in a carton of twelve are broken is 5%. Suppose that you are asked to calculate the probability that 2 of the next 20 cartons of eggs will have three or more broken eggs. Explain carefully what this question asks you to calculate. What is the answer? Explain why it would or would not be appropriate to use the normal approximation to the binomial to make this calculation.

7. An archer has a probability of hitting a bullseye of .6. What is the probability that the archer will hit 40 bullseyes in the next 70 shots? Explain why it would or would not be appropriate to use the normal approximation to the binomial in this case.

8. Suppose that for a very large sample of students, the mean time to finish a problem on a calculus exam has ranged from 8.5 to 17.25 minutes, with a uniform distribution. What is the probability that the next student will take between 12 and 15 minutes to do the problem?

Suppose instead that the mean time to finish the problem has been normally distributed, with a mean of 13.65 hours and a standard deviation of 3.75 hours. What is the probability that the next student will take between 12 and 15 minutes to do the problem?

9. An automobile manufacturer claims that the gas mileage on its car is 35 miles per gallon. A consumer group is skeptical and thinks that the mileage is less. State the null hypothesis and the alternative hypothesis.

10. The heights of adult women are normally distributed with a mean of 65 inches and a standard deviation of 2 inches. If Rachel is at the 99th percentile for height, how tall is she?

11. If you were shown a normal distribution where the mean is 50 and about 2.5 percent of the distribution is at 70 or above, what would you estimate the standard deviation of the distribution to be?

12. A study compares the SAT scores of students who take a particular prep course with students who do not take the course. The study reports that the 95 percent confidence interval for the difference between the mean test score of the prepped students minus the mean of the non-prepped students is (9,23). If ua is the true mean for prepped students and ub is the true mean for unprepped students, write an equation that describes the confidence interval in terms of ua and ub.

13. If someone told you that your z-score for cholesterol is 1.50, what does this say about your cholesterol?

14. A random sample of the costs of repair jobs at a muffler shop produces a mean of \$127.95 and a standard deviation of \$24.03. If the size of the sample is 40, what is a 90 percent confidence interval for the average cost of a repair at this shop?

15. Suppose that test scores on a chemistry final have a mean of 75 with a standard deviation of 12. Scores on a calculus final have a mean of 80 with a standard deviation of 8. If you had gotten an 81 in chemistry and an 84 in calculus, in which course did you do better relative to the rest of the class? Justify your answer.

16. In a test of the null hypothesis that the mean is 10, a sample produces a mean of 13.4 with a P-value of .017

Which of the following do we know?
(A) the probability of Type I error
(B) the probability of Type II error
(C) the significance level chosen for the test
(D) the power of the test

17. Loaves of bread at a bakery follow approximately a normal distribution. You find that 10 percent of loaves weigh less than 15.34 ounces and 20 percent of loaves weight more than 16.31 ounces. What are the mean and standard deviation for the distribution of the weights of the loaves of bread?

18. Suppose that you are asked to conduct an exit poll to determine the proportion of people who voted Democratic in a precinct. How large a sample would you need to take to have a margin of error no larger than .08?

19. A pain reliever claims that it has at least 200 milligrams of an active ingredient in each tablet. However, a sample is conducted and finds that out of 70 tablets, the mean is 194.3 milligrams with a standard deviation of 21 milligrams. What is the null hypothesis? What is the alternative hypothesis? What is the P-value?

20. Out of a sample of 100 college seniors, 34 indicated that they think they made the wrong choice of college. What is a 90 percent confidence interval for the percentage of all college seniors who think they made the wrong choice?