课程名称︰统计学一上
课程性质︰必修
课程教师︰蒋明晃
开课学院：管理学院
开课系所︰工管系
考试日期（年月日）︰104/12/31
考试时限（分钟）：3小时
试题 :
一、选择题 (共40 points)
1. An unbiased estimator of a population parameter is defined as:
a. an estimator whose variance is equal to one.
b. an estimator whose expected value is equal to zero.
c. an estimator whose variance goes to zero as the sample size goes to infinity.
d. None of these choices.
2. The letter α in the formula for constructing a confidence interval estimate
of the population mean is:
a. the level of confidence.
b. the probability that a particular confidence interval will contain the
population mean.
c. the area in the lower tail of the sampling distribution of the sample mean.
d. None of these choices.
3. In developing an interval estimate for a population mean, a sample of 50
observations was used. The interval estimate was 19.76±1.32. Had the sample
size been 200 instead of 50, the interval estimate would have been:
a. 19.76±.33
b. 19.76±.66
c. 19.76±5.28
d. None of these choices.
4. In developing an interval estimate for a population mean, the population
standard deviation σ was assumed to be 10. The interval estimate was 50.92±
2.14. Had σ equaled 20, the interval estimate would be
a. 60.92±2.14
b. 50.92±12.14
c. 101.84±4.28
d. 50.92±4.28
5. Suppose a 95% confidence interval for μ turns out to be (1,000, 2,100).
What does it mean to be 95% confidence?
a. In repeated sampling, the population parameter would fall in the resulting
interval 95% of the time.
b. 95% of the observations in the entire population fall in the given
interval.
c. 95% of the observations in the sample fall in the given interval.
d. None of these choices.
6. The sample size needed to estimate a population mean to within 10 units was
found to be 68. If the population standard deviation was 50, then the
confidence level used was:
a. 99%
b. 95%
c. 90%
d. None of these choices.
7. We cannot commit a Type I error when the:
a. null hypothesis is true.
b. level of significance is 0.10.
c. null hypothesis is false.
d. test is a two-tail test.
8. Suppose we wish to test H_0: μ=45 vs. H_1:μ>45. What will result if we
conclude that the mean is greater than 45 when the actual mean is 50?
a. We have made a Type I error.
b. We have made a Type II error.
c. We have made both a Type I error and a Type II error.
d. We have mad the correct decision.
9. The owner of a local Jazz Club has recently surveyed a random sample of
n=200 customers of the club. She would now like to determine whether or not
the mean age of her customers is over 30. If so, she plans to alter the
entertainment to appeal to an older crowd. If not, no entertainment changes
will be made. The appropriate hypotheses to test are:
a. H_0: μ=30 vs. H_1:μ<30.
b. H_0: μ=30 vs. H_1:μ>30.
_ _
c. H_0:X =30 vs. H_1:X <30.
_ _
d. H_0:X =30 vs. H_1:X >30.
10.In order to determine the p-value, which of the following is not needed?
a. The level of significance.
b. Whether the test is one-tail or two-tail.
c. The value of the test statistic.
d. All of these choices are true.
11.If a hypothesis is not rejected at the 0.10 level of significance, it:
a. must be rejected at the 0.05 level.
b. may be rejected at the 0.05 level.
c. will not be rejected at the 0.05 level.
d. must be rejected at the 0.025 level.
12.Using a confidence interval when conducting a two-tail test for μ, we do
not reject H_0 if the hypothesized value for μ:
a. is to the left of the lower confidence limit (LCL).
b. is to the right of the upper confidence limit (UCL).
c. falls between the LCL and UCL.
d. falls in the rejection region.
13.Suppose that in a certain hypothesis test the null hypothesis is rejected at
the .10 level; it is also rejected at the .05 level; however, it cannot be
rejected at the .01 level. The most accurate statement than can be made about
the p-value for this test is that:
a. p-value =0.01.
b. p-value =0.10.
c. 0.01< p-value <0.05.
d. 0.05< p-value <0.10.
14.We have created a 95% confidence interval for μ with the results (10, 25).
What conclusion will we make if we test H_0: μ=26 vs. H_1:μ≠26 at α=0.025?
a. Reject H_0 in favor of H_1
b. Accept H_0 in favor of H_1
c. Fail to reject H_0 in favor of H_1
d. We cannot tell from the information given.
15.Based on sample data, the 90% confidence interval limits for the population
mean are LCL = 170.86 and UCL = 195.42. If the 10% level of significance were
used in testing the hypotheses H_0: μ=201 vs. H_1:μ≠201, the null
hypothesis:
a. would be rejected.
b. would be accepted.
c. would fail to be rejected.
d. None of the above.
16.A random sample of 25 observations is selected from a normally distributed
population. The sample variance is 10. In the 95% confidence interval for the
population variance, the upper limit is:
a. 19.353
b. 17.331
c. 17.110
d. 6.097
17.In a hypothesis test for the population variance, the hypotheses are
H_0: σ^2=30 vs. H_1: σ^2<30. If the sample size is 20 and the test is being
carried out at the 5% level of significance, the null hypothesis is rejected
if :
a. X^2<30.144.
b. X^2>10.851.
c. X^2<10.117.
d. X^2>31.410.
18.In selecting the sample size to estimate the population proportion p, if we
have no knowledge of even the approximate values of the sample proportion p`,
we:
a. take another sample and estimate p`.
b. take two more samples and find the average of their p`.
c. let p`= 0.50.
d. let p` =0.95.
19.After calculating the sample size needed to estimate a population
proportion to within 0.04, your statistics professor told you the maximum
allowable error must be reduced to just .01. If the original calculation led
to a sample size of 800, the sample size will now have to be:
a. 800
b. 3,200
c. 6,400
d. 12,800
20.A survey claims that 9 out of 10 doctors recommend aspirin for their
patient with headaches. To test this claim against the alternative that the
actual proportion of doctors who recommend aspirin is less than 0.90, a
random sample of 100 doctors’ results in 83 who indicate that they recommend
aspirin. The value of the test statistic in this problem is approximately
equal to:
a. -1.67
b. -2.33
c. -1.86
d. -0.14
二、计算题：(共 60 points)
1. (16 points) A company claims that 10% of the users of a certain allergy
drug experience drowsiness. In clinical studies of this allergy drug, 81 of
the 900 subjects experienced drowsiness.
(1) (2 points) We want to test their claim and find out whether the actual
percentage is not 10%. State the appropriate null and alternative hypotheses.
(2) (4 points) Is there enough evidence at the 5% significance level to infer
that the company is correct?
(3) (4 points) Compute the p-value of the test.
(4) (4 points) Construct a 95% confidence interval estimate of the population
proportion of the users of this allergy drug who experience drowsiness?
(5) (2 points) Explain how to use this confidence interval to test the
hypotheses.
2. (12 points) Engineers who are in charge of the production of springs used
to make car seats are concerned about the variability in the length of the
springs. The springs are designed to be 500 mm long. When the springs are too
long, they will loosen and fail out. When they are too short, they will not
fit into the frames. The springs that are too long and too short must be
reworked at considerable additional cost. The engineers have calculated that
a standard deviation of 2 mm will result in an acceptable number of springs
that must be reworked. A random sample of 100 springs was measured, and the
sample variance is equal to 6.52.
(1) (6 points) Can we infer at the 5% significance level that the number of
springs requiring reworking is unacceptably large?
(2) (6 points) Suppose the engineers reduced the data so that springs that
were the correct length were recorded as 1, springs that were too long were
recorded as 2, and springs that were too short were recorded as 3. The
engineers found that there are 86 springs recorded as 1, 6 springs recorded
as 2, and 8 springs recorded as 3. Can we infer at the 10% significance level
that less than 90% of the springs are the correct length?
3. (16 points) A researcher wants to study the average lifetime of a certain
brand of light bulbs (in hours). In testing the hypotheses, H_0: μ=950 hours
vs. H_1:μ≠950 hours, a random sample of 25 light bulbs is drawn from a
normal population whose standard deviation is 200 hours.
(1) (5 points) Calculate β, the probability of a Type II error when μ=1000
and α=0.10.
(2) (4 points) Calculate the power of the test when μ=1000 and α=0.10.
(3) (4 points) Recalculate β if n is increased from 25 to 40.
(4) (3 points) Recalculate β if α is lowered from 0.10 to 0.05.
4. (9 points) A statistics professor would like to estimate a population mean
to within 40 units with 99% confidence given that the population standard
deviation is 200.
(1) (3 points) What sample size should be used?
(2) (2 points) What sample size should be used if the standard deviation is
changed to 50?
(3) (2 points) What sample size should be used if using a 95% confidence
level?
(4) (2 points) What sample size should be used if we wish to estimate the
population mean to within 10 units?
5. (7 points) An accountant was performing an audit for a tractor dealership.
An auditor wants to examine the monetary error made by the purchasing order
department in the month of July. He decided to randomly sample 100 of the 925
purchase orders for the month of July, and found the amount of error in each
_
one. The statistics for this sample were: x =$6.0 and s=$17.012.
(1) (4 points) Estimate with 95% confidence the average amount of error per
purchase order for the entire month of July. (Hint: Should the finite
population correction factor needs to be used?)
(2) (3 points) Estimate with 95% confidence the total amount of monetary
error for the month of July.
Solution
选择题
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