AP Statistics Test 6
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1 AP Statistics Test 6 Name: Date: Period: ffl If X 1 ;X 2 ;:::;X n are random variables, then E(X 1 +X 2 + +X n )=E(X 1 )+ E(X 2 )+ + E(X n ): ffl If X 1 ;X 2 ;:::;X n are independent random variables, then Var(X 1 +X X n )=Var(X 1 )+Var(X 2 )+ +Var(X n ): ffl If X 1 ;X 2 ;:::;X n are independent random variables, each having mean μ and variance ff 2, and if X = X 1 + X X n ; then μ n X = μ and ff X = ff p. n ffl 100(1 ff)% Confidence Intervals: x ± z ff=2 s x p n (large samples) x ± t ff=2 s x p n (small samples, underlying distribution roughly normal) È bp(1 bp) È bp ± z ff=2 p (large sample proportions); p(1 p)» 1=2 with equality when p n =1=2. normalcdf( z ff=2 ;z ff=2 )=1 ff ffl Useful z-values: z :05 = 1:645 z :025 =1:960 z :005 =2:576. cdf( t ff=2 ;t ff=2 ;n 1) = 1 ff: Part 1, Multiple-Choice Questions 50 points. 1. Suppose that we have a random variable X having mean μ = 2:3 and standard deviation ff X = 1:44. If we take samples of X of size 120, then the parameters of the sample mean X are given by (A) μ X =2:3; ff X = :012 (B) μ X =2:09; ff X = :131 (C) μ X = :019; ff X = :012 (D) μ X =2:3; ff X = :131 (E) μ X =2:3; ff X =1:44
2 2. Which of the following statements are true? I. The mean of the set of sample means varies inversely as the square root of the size of the sample. II. The variance of the sample means varies inversely as the size of the sample. III. The standard deviation of the sample means varies inversely as the square root of the size of the sample. (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III 3. A 95% confidence interval for the mean μ of a population is computed from a random sample and found to be 9 ± 3. We may conclude that (A) There is a 95% probability that μ is between 6 and 12 (B) If we took many additional random samples and computed the resulting 95% confidence intervals, then roughly 95% of these intervals would contain μ. (C) There is a 95% probability that the true mean is 9, and there is a 95% probability that the true margin of error is 3. (D) All of the above are true. (E) None of the above is true. 4. Suppose that we have computed a 95% confidence interval (from a large sample) for the mean μ X of a random variable X. If we increase the confidence level to 99%, then the width of the confidence interval will (A) increase by about 31%. (B) increase by about 19%. (C) decrease b y about 31%. (D) decrease by about 19%. (E) not change. 5. Suppose that we have a population from which a sample of size 200 is taken, with the result that x = 4:23 and s x = 0:785. If we compute a confidence interval with a 0.06 margin of error, then the confidence level is approximately (A) 66% (B) 72% (C) 97% (D) 90% (E) 95%
3 6. An agricultural researcher plants 25 plots with a new variety of corn. The average yield for these plots is x = 150 bushels per acre. Assume that the yield per acre for the new variety of corn roughly follows a normal distribution with unknown mean μ and standard deviation ff =10bushels per acre. A 90% confidence interval for μ is (A) 150 ± 2:00 (B) 150 ± 3:29 (C) 150 ± 3:42 (D) 150 ± 3:92 (E) 150 ± 32:9 7. Suppose that we are trying to determine the percentage of students that have math addiction. How large a sample should be taken in order to guarantee a margin of error of no more than ±3% at a confidence level of 95%? (A) 6 (B) 33 (C) 534 (D) 752 (E) In general, how does tripling the sample size change the margin of error? (A) It triples the margin of error. (B) It divides the margin of error by 3. (C) It multiplies the margin of error by (D) It divides the margin of error by (E) This question cannot be answered without knowing the original sample size. 9. The following represents the number of printed characters (in millions of characters) before each of 15 printers failed A 95% confidence interval for the mean number of printed characters (in millions) before printer failure is given by (A) 1:16» μ» 1:32 (B) 1:15» μ» 1:33 (C) 1:14» μ» 1:34 (D) 1:13» μ» 1:35 (E) 1:12» μ» 1:36
4 10. The following histogram represents 14 samples of the random variable X: f4:2; 4:3; 4:4; 4:8; 6:2; 6:7; 8:1; 8:2; 9:5; 10:2; 10:6; 11:4; 11:4; 11:6g. Based on the above, which of the following statements are correct? I. It doesn t appear that the sample was drawn from a normal distribution. II. Using large-sample techniques based on the z-distribution for finding a 95% confidence interval for μ X is probably not appropriate. III. Using small-sample techniques based on the t-distribution for finding a 95% confidence for μ X is probably not appropriate. (A) I only (B) II only (C) III only (D) I and II (E) I, II, and III
5 Part 2, Free-Response Questions 50 points. 11. Suppose that 10 AP Statistics students each take 40 samples of the random normal variable X, via the TI command randnorm(3:2; 0:13; 40)! L 1 : From their samples, each computes a 95% confidence interval using the TI command ZInterval. (a) (3 points) What is the probability that all 10 of the students confidence intervals will contain the true mean (namely, 3.2)? Show your work, explaining your reasoning. (b) (3 points) What is the probability that at least two of the students confidence intervals will fail to contain the true mean? Show your work, explaining your reasoning. (c) (3 points) What is the expected number of students whose confidence intervals will contain the true mean? Explain.
6 12. A random sample of 1,243 adult U.S. citizens were asked the following question: Based on what you know about the Social Security System today, what would you like Congress and the President to do during this next year? The response choices and the percentages selecting them are shown to the right. Completely overhaul the system 19% Make some major changes 39% Make some minor adjustments 30% Leave the system the way it is now 11% No opinion 1% (a) (3 points) Find a 95% confidence interval for the proportion of all United States adults who would respond Make some major changes to the question. (b) (3 points) Interpret the confidence interval in the context of the question. (c) (3 points) Interpret the confidence level in the context of the question.
7 13. Researchers are planning a study of a certain mild illness. They will select a random sample of patients who are ages 35 to 54 and see if they contract the illness in the next year. The researchers are interested in estimating the proportions of men and of women who are likely to develop the illness in each of 4 age groups: 35 39, 40 44, 45 49, and The researchers included 2,000 patients in the study, with the results tabulated to the right. Male Female Age Group (a) (8 points) Suppose that at the end of the study, 10 percent of the females in the age group contracted the illness. Calculate a 95% confidence interval to estimate the population proportion of females in this age group that contracted the illness. Interpret the confidence interval in the context of this situation. Interpret the confidence level of 95%. (b) (5 points) Suppose that at the end of the study, 10 percent of the males in the age group contracted the illness. The corresponding 95% confidence interval to estimate the population proportion of males in this age group that contracted the illness is (0:061; 0:139). Note that this interval and the interval you computed in (a) are of different lengths even though the two sample proportions were identical. What would be an alternative way to allocate a sample of 2,000 subjects so that the 95% percent confidence interval widths for all male age groups (i.e., for all 8 groups) would be the same when the sample proportions are the same? Justify your answer.
8 14. (Adapted from our text, McClave and Sinich, Ninth Edition) The Austrailan Journal of Zoology (Vol. 32, 1995) reported on a study of the diets and water requirements of spinifex pigeons. Sixteen pigeons were captured in the desert and the crop (i.e., stomach) contents of each examined. The accompanying table reports the weight (in grams) of dry seed in the crop of each pigeon. The table below reveals the data (a) (3 points) Compute a 99% confidence interval for the mean crop of desert pigeons, explaining any assumptions you have made in validating your method. (b) (3 points) Explain the meaning of your interval in the context of this problem. (c) (3 points) Explain the meaning of the confidence level in the context of this problem.
9 15. A simple random sample of 40 inner city gas stations shows a mean price for regular unleaded gasoline to $3.27 with a standard deviation of $0.14, while a simple random sample of 120 suburban stations shows a mean price of $3.03 with a standard deviation of $0.19. (a) (4 points) Construct 95% confidence intervals for the mean price of regular gasoline in inner city and in suburban stations. (b) (3 points) The confidence interval for the inner city stations is wider than the interval for the suburban stations even through the standard deviation for inner city stations is less than that for suburban stations. Explain why this happened. (c) (3 points) Based on your answer to part (a), are you confident that the mean price of inner city gasoline is less than $3.15? Explain.
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