Multiple-Choice Questions

Transcription

1 AP Statistics Testbank 6 Formulas: If,, , n are random variables, then E ( n ) = E( 1 ) + E( 2 ) + + E( n ). If,, , n independent random variables, then Var ) = Var( ) + Var( ) + + Var( ). ( 1 2 n 1 2 n µ = µ, σ = σ / n 100(1 α) % Confidence Intervals: s ± zα / 2 (large samples) normalcdf( z α / 2, zα / 2 ) = 1 α n s ± tα / 2 (small samples; underlying distribution roughly normal) n cdf( t α t, n-1) = 1 α ^ p / 2, α / 2 ^ ^ p (1 p ) ± z α / 2 (large samples; proportions) p ( 1 p) 1/ 2 with equality when p = ½. n Multiple-Choice Questions 1) Suppose that the random variable has the distribution given below: P() Confidence Intervals : 90% : z 95% : z 99% : z Now suppose that we are to take random samples of size n=50 from the population. Then the parameters of the sample mean are given by a) µ = 15, σ =.521 b) µ = 15, σ =.721 c) µ = 14.55, σ =.517 d) µ = 14.55, σ =.719 e) µ = 2.06, σ =.719 2) Suppose that we have a random variable having mean µ = 2. 3, and standard deviation σ = If we take samples of size 120, then the parameters of the sample mean are given by a) µ = 2.3, σ =.012 b) µ =.209, σ =.131 c) µ =.019, σ =.012 d) µ = 2.3, σ =.131 e) µ = 2.3, σ = 1.44

2 2 3) Suppose that is a random variable with mean µ and variance σ. If we take samples of size n, what can µ we say about the statistic X =? σ / n (I) X is approimately normally distributed. (II) X has standard deviation σ / n. (III) X has mean 0. a) I only b) II only c) III only d) I and II e) I and III 4) 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 samples. (II) The variance of the sample mean varies directly as the size of the samples and inversely as the variance of the original population. (III) The standard deviation of the sample mean varies directly as the standard deviation of the original population and inversely as the square root of the size of the samples. a) I only b) II only c) III only d) I and II e) I and III 5) Suppose that is a normally-distributed random variable with mean µ. If we take samples of size n, what µ can we say about the statistic X =? s / n (I) X is approimately normally distributed with mean 0. (II) X has the t-distribution with n-1 degrees of freedom. (III) If n is large, X is approimately normally distributed with mean 0. a) I only b) II only c) I and II d) I and III e) II and III 6) Suppose we have a population with proportion p. From this population we select a sample of size n and in ^ ^ this sample we measure the proportion p. The standard deviation of p is a) p( 1 p) b) p ( 1 p) / n c) ^ ^ p (1 p ) d) ^ ^ p(1 p ) / n e) Impossible to calculate from the information given.

3 7) 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) there is a 95% probability that the true mean is 9 and there is a 95% chance that the true margin of error is 3. c) if we took many additional random samples and from each computed a 95% confidence interval for µ, approimately 95% of these intervals would contain µ. d) all of the above. e) none of the above. 8) Suppose that we have computed a 95% confidence interval (from a large sample) for the mean µ of a random variable. 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 by about 31% d) decrease by about 19% e) not change. 9) Suppose that we have computed a 95% confidence interval (from a large sample) for the mean µ of a random variable. If we decrease the confidence level to 90%, then the width of the confidence interval will a) increase by about 19% b) increase by about 31% c) decrease by about 16% d) decrease by about 24% e) not change. 10) Suppose that we have a population from which a sample of size 200 is taken, with the result that = and s = If we compute a confidence interval with a margin of error of 0.06, then the confidence level is approimately a) 66% b) 72% c) 97% d) 90% e) 95% 11) Suppose that we take a sample of size 120 from a population, measuring = 6. 2 and = 2.3. A 90% confidence interval for the mean µ is a) 5.9 µ 6. 5 b) 5.8 µ 6. 6 c) 5.7 µ 6. 7 d) 5.6 µ 6. 8 e) 5.5 µ 6. 9 s

4 12) Suppose that we take a sample of size 120 from a population, measuring = 6. 2 and s = 2.3. The confidence interval 6.2 ± 0.2 would represent a confidence level of approimately a) 55% b) 66% c) 90% d) 95% e) 99% 13) Suppose that we take a sample of size 20 from an approimately normally-distributed population, measuring = 6.25 and s = A 95% confidence interval for the mean µ is a) 5.40 µ b) 5.36 µ c) 5.17 µ d) 5.23 µ e) 4.77 µ ) An agricultural researcher plants 25 plots with a new variety of corn. The average yield for these plots is = 150 bushels per acre. Assume that the yield per acre for the new variety of corn follows a normal distribution with unknown mean µ and standard deviation σ = 10 bushels per acre. A 90% confidence interval for µ is a) 150 ± b) 150 ± c) 150 ± d) 150 ± e) 150 ± ) A recent CNN report is that 43% of high school students have a genuine dislike of statistics. Which of the following best describes what is meant by the poll having a margin of error of 5%? a) It is likely that the true proportion of high school students having a dislike of statistics is between 38% and 48%. b) 5% of the students refused to participate in this poll. c) Between 38% and 48% of those surveyed epressed a dislike of statistics. d) There is a.05 probability that the 43% result is in error. e) If similar size polls were repeatedly taken, they would be wrong about 5% of the time. 16) 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) 1068

5 17) 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 with knowing the original sample size. 18) A confidence interval based on a random sample of n families for the mean grocery ependiture has been determined. Which of the following will result in a smaller margin of error? (I) A smaller confidence level (II) A smaller sample standard deviation (III) A smaller sample size. a) II only b) I and II c) I and III d) II and III e) I, II, and III 19) An insurance actuary studying the average length of stay for hospital patients determines that for a 95% confidence level estimate of the average length of stay to be within ± days, 100 patients records need to be eamined. How many patient records should be eamined in order to reduce the margin of error to within ± 0.25 days and still retain a confidence of 95%? a) 25 b) 50 c) 200 d) 400 e) There is not enough information given. 20) 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 b) 1.15 µ c) 1.14 µ d) 1.13 µ e) 1.12 µ 1. 36

6 21) Suppose that is a random variable and that we select a sample of size 150, with the result that = and the s = If we estimate the population mean µ with a confidence interval having margin of error ±.935, what, approimately, is the corresponding confidence level? a) 50% b) 60% c) 70% d) 80% e) 90% 22) The following histogram represents the 14 data values {4.2, 4.3, 4.4, 4.8, 6.2, 6.7, 8.1, 8.2, 9.5, 10.2, 10.6, 11.4, 11.4, 11.6}: Based on the above, which of the following statements are correct? (I) (II) (III) It doesn t appear that the sample was drawn from a normal distribution. Using large-sample techniques based on the z-distribution for finding a 95% confidence interval for µ is probably not appropriate. Using small-sample techniques based on the t-distribution for finding a 95% confidence interval for µ is probably not appropriate. a) I only b) II only c) III only d) I and II e) I, II, and III

7 Free-Response Questions 23) Suppose that the SAS varsity basketball team has a roster of 12 players, whose average number of points per game and standard deviations are given. Player Name Point Avg. St. Dev Brian Haonan Henry Aaron Ale Billy Tory Joey Howard Gordon Vikas Dong Whi a) What is the average number of points scored by the SAS varsity basketball team? Avg: b) What is the standard deviation of the number of points scored by the SAS varsity basketball team? Standard Deviation: c) Assuming that the distribution of the total number of points scored each game by the SAS varsity basketball team is approimately normal, approimate the probability that in the net game the team will score 80 or more points. Probability:

8 24) The following is the result of a survey of 100 SAS high school students regarding the possibility of implementing a dress code: Preference Favor Dress Code Oppose Dress Code Total Compute a 95% confidence interval for the overall proportion of SAS high school students favoring a dress code. Ans: 25) A random sample of 1243 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 net year? The response choices and the percentages selecting them are shown below. 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) Find a 95% confidence interval for the proportion of all United States adults who would respond Make some major changes to the question. 95% Confidence Interval: b) Give an interpretation of the confidence interval and give an interpretation of the confidence level.

9 26) (Adapted from our tet, McClave and Sinich, Ninth Edition) The Australian Journal of Zoology (Vol. 43, 1995) reported on a study of the diets and water requirements of spinife pigeons. Siteen pigeons were captured in the desert and the crop (i.e., stomach) contents of each eamined. The accompanying table reports the weight (in grams) of dry seed in the crop of each pigeon. Use the accompanying TI-83 printout to find a 99% confidence interval for the average weight of dry seeds in the crops of spinife pigeons inhabiting the Western Australian desert. Interpret the result Var Stats = = = S = σ = % Confidence Interval: 27) Suppose that we wish to find a confidence interval for the proportion of satisfied owners of the Volkswagon Polo automobile. Compute the conservative estimate of the number n of owners that must be surveyed in order to obtain a margin of error no more than 1.5% at a confidence level of 90%. n

10 28) A simple random sample of 40 inner city gas stations shows a mean price for regular unleaded gasoline to be $1.45 with a standard deviation of $0.05, while a simple random sample of 120 suburban stations shows a mean of $1.38 with a standard deviation of $0.08. a) Construct 95% confidence intervals for the mean price of regular gasoline in inner city and in suburban stations. 95% Confidence Interval (inner city) 95% Confidence Interval (suburban) b) The confidence interval for the inner city stations is wider than the interval for the suburban stations even though the standard deviation for inner city stations is less than that for suburban stations. Eplain why this happened. c) Based on your answer in part (a), are you confident that the mean price of inner city gasoline is less than $1.07? Eplain.