Statistics (I)
Final Exam
Jan. 11, 2002
解答：熬夜怎么办d d a d a , c c d d c
Multiple 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 lected 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 reprentation of the population from which it was collected is referred to as a(n) ____ sample.
a. Biad b. Empirical c. Statistic d. Inferential
4. Suppo a 95% confidence interval for μturns out to be (1,000, 2,100). Bad 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 increa in the sample size will lead to a ______ interval.
a. Narrower b. Wider c. Less significant d. Biad
6. The sample size formulas are uful for ___________.
a. Calculating a value of the test statistic when testing n
b. 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. Suppo we wish to test H0: μ= 47 vs Ha: μ> 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 hypothesis
b. The test statistic
c. The sampling distribution of the test statistic
d. All of above
10. We have created a 95% confidence interval for µ with the result (10, 15). What conclu
sion will we make if we test H0: μ= 16 vs Ha: μ> 16 at α= 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 Questions
1. 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 bank’s accounts to asss the reliability of the mean balance reported by the bank. If the bank’s 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, )
感恩资助P(<1072) = P() = P(Z<2.4145)=0.992
2. Let X and Y be independent random variables with E(X)=3, E(Y)=5, Var(X)=Var(Y)=σ2, find the value of M which makes M(X2 - Y2谢干权)+Y2 be an unbiad estimate of σ荷花池2. (12%)
答：須使M(X2 - Y2)+Y2之期望值等於σ2
E(X)=3, E(Y)=5 Var(X)=Var(Y)=σ2
⇒E{ M(X2 - Y2)+Y2} = M⋅E(X2) + (1-M)⋅E(Y2)
其中，Var(X)=E(X2)-[E(X)]2 ⇒ σ2=E(X2)-9 ⇒ E(X2)= σ2+9
Var(Y)=E(Y热情的意思2)-[E(Y)]2 ⇒ σ小熊玩具2=E(Y2)-25⇒ E(Y2)= σ2+25
E{ M(X2 - Y2)+Y2} = M⋅E(X2) + (1-M)⋅E(Y2) = σ2 ( 不偏估計元)
⇒ M⋅[σ2+9] + (1-M)⋅[ σ2+25] = σ2
⇒ σ2 -16M +红苹果图片 25 = σ2 ⇒ 16M = 25 ⇒ M = 25/16
3. A normal population has unknown variance, σ2. If (1.48, 8.5) is a 95% confidence interval forσ2 bad on a random sample of size 12, find the value of . (10%)
答：σ2=(1.48, 8.5) ⇔ 1.48< σ2< 8.5
⇒ 推想σ2信賴區間之計算方法 ⇒ χ2統計量=~ χ2(n-1)
n=12, α=0.05 ⇒信賴區間之建立：
八分饱⇒
⇒⇒⇒ =8.5⨯3.816=32.436
或 ⇒ ⇒=1.48⨯21.92=32.4416
