Statistics-(I).doc

Statistics (I)Final ExamJan. 11, 2002 解答：d d a d a , c c d d cMultiple Choice Questions (30%)1. Which of the following is not one of the sampling distribution of the sample mean?a. The mean of the sampling distribution is equal to the population mean.b. The standard deviation of the sampling distribution is equal to the standard deviation of the population divided by the square root of n.c. The shape of the sampling distribution is approximately normal if the sample size is sufficiently large.d. All of the above are properties of the sampling distribution of the sample mean.2. Why is the Central Limit Theorem so important to the study of sampling distributions?a. It allows us to disregard the size of the sample selected when the population is approximately normal.b. It allows us to disregard the shape of the sampling distribution when the size of population is large.c. It allows us to disregard the size of the population we are sampling from.d. It allows us to disregard the shape of population when n is large.3. A sample that does not provide a good representation of the population from which it was collected is referred to as a(n) _ sample.a. Biased b. Empirical c. Statistic d. Inferential4. Suppose a 95% confidence interval for turns out to be (1,000, 2,100). Based on the interval, do you believe the average is equal to 2,200?a. Yes, and I am 95% sure of it.b. Yes, and I am 100% sure of it.c. No, and I am 95% sure of it.d. 1,200 is a possible value for the mean.5. In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a _ interval.a. Narrower b. Wider c. Less significant d. Biased6. The sample size formulas are useful for _.a. Calculating a value of the test statistic when testing nb. Calculating a p-value for a large sample test of n.c. Calculating the sample size to estimate a parameter to within a specified bound.d. Comparing different confidence intervals constructed using different sample sizes.7. Suppose we wish to test H0: = 47 vs Ha: 47. What will result if we conclude that the mean is greater than 47when its true value is really 52?a. We have made a Type II error.b. We have made a Type I error.c. We have made a correct decision.d. None of the above are correct.8. Which of the following statements is Not true about the level of significance in a test of hypothesis?a. The larger the level of significance, the more likely you are to reject the null hypothesis. b. The level of significance is the maximum risk we are willing to accept in making a Type I error.c. The level of significance is also called the alpha level.d. The level of significance is another name for a p-value.9. Which part of the test of hypothesis procedure determines the value of the p-value?a. The alternative hypothesisb. The test statisticc. The sampling distribution of the test statisticd. All of above10. We have created a 95% confidence interval for with the result (10, 15). What conclusion will we make if we test H0: = 16 vs Ha: 16 at = 0.05?a. Reject H0 in favor of Ha.b. Accept H0 in favor of H0.c. Fail to reject H0.d. We cannot tell what our decision will be with the information given.Essay Questions1. A local bank reported to the federal government that its 5,246 savings accounts have a mean balance of $1,000 and a standard deviation of $240. Government auditors have asked to randomly sample 64 of the banks accounts to assess the reliability of the mean balance reported by the bank. If the banks information is correct, find the probability that the sample mean balance would be less than $1,072. (8%)答：N=5246(有限母體隨機樣本並非完全獨立變異數要做修正)n=64 N(1000, )P(1072) = P() = P(Z2.4145)=0.992 2. Let X and Y be independent random variables with E(X)=3, E(Y)=5, Var(X)=Var(Y)=2, find the value of M which makes M(X2 - Y2)+Y2 be an unbiased estimate of 2. (12%)答：須使M(X2 - Y2)+Y2之期望值等於s2 Q E(X)=3, E(Y)=5 Var(X)=Var(Y)=s2 E M(X2 - Y2)+Y2 = ME(X2) + (1-M)E(Y2) 其中，Var(X)=E(X2)-E(X)2 s2=E(X2)-9 E(X2)= s2+9Var(Y)=E(Y2)-E(Y)2 s2=E(Y2)-25 E(Y2)= s2+25E M(X2 - Y2)+Y2 = ME(X2) + (1-M)E(Y2) = s2 (Q不偏估計元) Ms2+9 + (1-M) s2+25 = s2 s2 -16M + 25 = s2 16M = 25 M = 25/163. A normal population has unknown variance, 2. If (1.48, 8.5) is a 95% confidence interval for2 based