Statistics Midterm Test 108/2 (A 卷)

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[Notes]The attached probability value is the probability in upper tail of 𝔃 𝐨𝐫 𝓽 distribution
𝔃𝟎.𝟑𝟖𝟐𝟏 = 𝟎. 𝟑 𝔃𝟎.𝟑𝟕𝟎𝟕 = 𝟎. 𝟑𝟑 𝔃𝟎.𝟐𝟐𝟑𝟔 = 𝟎. 𝟕𝟔 𝔃𝟎.𝟏𝟐𝟑 = 𝟏. 𝟏𝟔 𝔃𝟎.𝟏𝟏𝟓𝟏 = 𝟏. 𝟐 𝔃𝟎.𝟏𝟎𝟑𝟖 = 𝟏. 𝟐𝟔 𝔃𝟎.𝟏𝟎𝟐 = 𝟏. 𝟐𝟕
𝔃𝟎.𝟏𝟎𝟎𝟑 = 𝟏. 𝟐𝟖 𝔃𝟎.𝟏𝟎 = 𝟏. 𝟐𝟖𝟐 𝔃𝟎.𝟎𝟗𝟑𝟒 = 𝟏. 𝟑𝟐 𝔃𝟎.𝟎𝟗𝟏𝟖 = 𝟏. 𝟑𝟑 𝔃𝟎.𝟎𝟖𝟔𝟗 = 𝟏. 𝟑𝟔 𝔃𝟎.𝟎𝟖𝟓𝟑 = 𝟏. 𝟑𝟕
𝔃𝟎.𝟎𝟓 = 𝟏. 𝟔𝟒𝟓 𝔃𝟎.𝟎𝟒𝟕𝟓 = 𝟏. 𝟔𝟕 𝔃𝟎.𝟎𝟒𝟔𝟓 = 𝟏. 𝟔𝟖 𝔃𝟎.𝟎𝟐𝟕𝟒 = 𝟏. 𝟗𝟐 𝔃𝟎.𝟎𝟐𝟔𝟖 = 𝟏. 𝟗𝟑 𝔃𝟎.𝟎𝟐𝟔𝟔 = 𝟏. 𝟗𝟓
𝔃𝟎.𝟎𝟐𝟓 = 𝟏. 𝟗𝟔 𝔃𝟎.𝟎𝟐𝟒𝟒 = 𝟏. 𝟗𝟕 𝔃𝟎.𝟎𝟏𝟖𝟑 = 𝟐. 𝟎𝟗 𝔃𝟎.𝟎𝟏𝟕𝟒 = 𝟐. 𝟏𝟏 𝔃𝟎.𝟎𝟏𝟔𝟔 = 𝟐. 𝟏𝟑 𝔃𝟎.𝟎𝟏𝟐𝟐 = 𝟐. 𝟐𝟓
𝔃𝟎.𝟎𝟏 = 𝟐. 𝟑𝟐𝟔 𝔃𝟎.𝟎𝟎𝟖𝟐 = 𝟐. 𝟒 𝔃𝟎.𝟎𝟎𝟓𝟕 = 𝟐. 𝟓𝟑 𝔃𝟎.𝟎𝟎𝟓 = 𝟐. 𝟓𝟕𝟔 𝔃𝟎.𝟎𝟎𝟐𝟗 = 𝟐. 𝟕𝟔 𝔃𝟎.𝟎𝟎𝟎𝟖 = 𝟑. 𝟏𝟔
𝓽𝟎.𝟐 (𝟗) = 𝟎. 𝟖𝟖𝟑 𝓽𝟎.𝟏 (𝟗) = 𝟏. 𝟑𝟖𝟑 𝓽𝟎.𝟎𝟓 (𝟗) = 𝟏. 𝟖𝟑𝟑 𝓽𝟎.𝟎𝟐𝟓 (𝟗) = 𝟐. 𝟐𝟔𝟐 𝓽𝟎.𝟎𝟎𝟓 (𝟗) = 𝟑. 𝟐𝟓𝟎
𝓽𝟎.𝟐 (𝟏𝟎) = 𝟎. 𝟖𝟕𝟗 𝓽𝟎.𝟏 (𝟏𝟎) = 𝟏. 𝟑𝟕𝟐 𝓽𝟎.𝟎𝟓 (𝟏𝟎) = 𝟏. 𝟖𝟏𝟐 𝓽𝟎.𝟎𝟐𝟓 (𝟏𝟎) = 𝟐. 𝟐𝟐𝟖 𝓽𝟎.𝟎𝟎𝟓 (𝟏𝟎) = 𝟑. 𝟏𝟔𝟗
𝓽𝟎.𝟏𝟎 (𝟏𝟏) = 𝟏. 𝟑𝟔𝟑 𝓽𝟎.𝟎𝟓 (𝟏𝟏) = 𝟏. 𝟕𝟗𝟔 𝓽𝟎.𝟎𝟐𝟓 (𝟏𝟏) = 𝟐. 𝟐𝟎𝟏 𝓽𝟎.𝟎𝟏 (𝟏𝟏) = 𝟐. 𝟕𝟏𝟖 𝓽𝟎.𝟏𝟎 (𝟏𝟐) = 𝟏. 𝟑𝟓𝟔
𝓽𝟎.𝟎𝟓 (𝟏𝟐) = 𝟏. 𝟕𝟖𝟐 𝓽𝟎.𝟎𝟐𝟓 (𝟏𝟐) = 𝟐. 𝟏𝟕𝟗 𝓽𝟎.𝟎𝟏 (𝟏𝟐) = 𝟐. 𝟔𝟖𝟏 𝓽𝟎.𝟎𝟓 (𝟏𝟔) = 𝟏. 𝟕𝟒𝟔 𝓽𝟎.𝟎𝟐𝟓 (𝟏𝟔) = 𝟐. 𝟏𝟐𝟎
𝓽𝟎.𝟎𝟏 (𝟏𝟔) = 𝟐. 𝟓𝟖𝟑 𝓽𝟎.𝟎𝟎𝟓 (𝟏𝟔) = 𝟐. 𝟗𝟐𝟏 𝓽𝟎.𝟐 (𝟏𝟕) = 𝟎. 𝟖𝟔𝟑 𝓽𝟎.𝟏 (𝟏𝟕) = 𝟏. 𝟑𝟑𝟑 𝓽𝟎.𝟎𝟓 (𝟏𝟕) = 𝟏. 𝟕𝟒𝟎
𝓽𝟎.𝟎𝟐𝟓 (𝟏𝟕) = 𝟐. 𝟏𝟏𝟎 𝓽𝟎.𝟎𝟏 (𝟏𝟕) = 𝟐. 𝟓𝟔𝟕 𝓽𝟎.𝟎𝟎𝟓 (𝟏𝟕) = 𝟐. 𝟖𝟗𝟖 𝓽𝟎.𝟐 (𝟏𝟖) = 𝟎. 𝟖𝟔𝟓 𝓽𝟎.𝟏 (𝟏𝟖) = 𝟏. 𝟑𝟑𝟎
𝓽𝟎.𝟎𝟓 (𝟏𝟖) = 𝟏. 𝟕𝟑𝟒 𝓽𝟎𝟎𝟐𝟓 (𝟏𝟖) = 𝟐. 𝟏𝟎𝟏 𝓽𝟎.𝟎𝟓 (𝟐𝟎) = 𝟏. 𝟕𝟐𝟓 𝓽𝟎.𝟎𝟐𝟓 (𝟐𝟎) = 𝟐. 𝟎𝟖𝟔 𝓽𝟎.𝟎𝟏 (𝟐𝟎) = 𝟐. 𝟓𝟐𝟖
𝓽𝟎.𝟎𝟓 (𝟕𝟒) = 𝟏. 𝟔𝟔𝟔 𝓽𝟎𝟎𝟐𝟓 (𝟕𝟒) = 𝟏. 𝟗𝟗𝟑 𝓽𝟎.𝟎𝟏 (𝟕𝟒) = 𝟐. 𝟑𝟕𝟖 𝓽𝟎.𝟎𝟎𝟓 (𝟕𝟒) = 𝟐. 𝟔𝟒𝟒 𝓽𝟎.𝟎𝟓 (𝟕𝟓) = 𝟏. 𝟔𝟔𝟓

PART I. MULTIPLE CHOICE QUESTIONS (2% for each)

1. The power curve provides the probability of
(A) correctly rejecting the alternative hypothesis (B) incorrectly accepting the null hypothesis
(C) correctly rejecting the null hypothesis (D) correctly accepting the null hypothesis
2. When developing an interval estimate for the difference between two sample means, with sample sizes of 𝑛1 and
𝑛2 ,
(A) 𝑛1 must be smaller than 𝑛2 (B) 𝑛1 and 𝑛2 can be of different sizes
(C) 𝑛1 must be larger than 𝑛2 (D) 𝑛1 must be equal to 𝑛2
3. What is the probability of making a Type II error if the null hypothesis is actually true?
(A) 𝛼 (B) 1 (C) 0.5 (D) 0
4. In what type of test is the variable of interest the difference between the values of the observations rather than the
observations themselves?
(A) A test for the difference between the means of 2 related populations.
(B) A test for the equality of variances from 2 independent populations.
(C) A test for the difference between the means of 2 independent populations.
(D) A test for of equality of variances from two related populations.
5. It is possible to directly compare the results of a confidence interval estimate to the results obtained by testing a
null hypothesis if
(A) a lower-tailed test for 𝜇 is used (B) a upper-tailed test for 𝜇 is used
(C) a two-tailed test for 𝜇 is used (D) None of the previous statements is true
6. Which condition must be met to conduct a test for the difference in two sample means using a z-statistic?
(A) The populations must be normal (B) The samples are dependent
(C) The data must be at least of nominal scale (D) The two population standard deviations must be known
 統計學(二) 0081 統計學 (A 卷) 本試題共 4 頁第 2 頁
7. In hypothesis testing, the critical value is
(A) the same as the p-value (B) a number that establishes the boundary of the rejection region
(C) the probability of a Type I error (D) the probability of a Type II error
8. For a given level of significance, if the sample size n is increased, the probability of a Type II error
(A) will remain the same (B) cannot be determined (C) will decrease (D) will increase
9. The t test for the difference between the means of two independent populations assumes that the respective:
(A) populations are approximately normal (B) sample sizes are equal
(C) population variances are known (D) sample variances are equal
10. The location of the rejection region for the null hypothesis is determined by
(A) the sample size 𝑛 (B) the sign in the null hypothesis
(C) the size of 𝛼 (D) the sign in the alternative hypothesis

PART II. MULTIPLE CHOICE QUESTIONS (4% for each)

Exhibit 1: (題組 11~ 12)
There are 75 customers sampled from a store. Their average waiting time for checking out is 3.1 minutes with a
standard deviation of 0.5 minutes. We want to determine whether or not the mean waiting time of all customers is
significantly more than 3 minutes.
11. Refer to Exhibit 1. What is the value of test statistic? What is the 𝒑-value for the hypothesis?
(A) 3.464; < 0.005 (B) 3.464; < 0.01 (C) 1.5; 0.05~0.10 (D) 1.732; 0.05~0.10 (E) 1.732; 0.025~0.05
12. Refer to Exhibit 1. Use 𝛼 = 0.05.What are critical value and it can be concluded that the mean of the population
is?
(A)1.666; significantly greater than 3 (B)1.666; significantly less than 3 (C)1.645; significantly less than 3
(D)1.645; significantly greater than 3 (E)1.665; not significantly greater than 3
Exhibit 2: (題組 13 ~14)
The following data are from matched samples taken from two populations.
Population 1 35 29 28 34 30 31 25 20 46 27
Population 2 25 18 20 22 26 36 22 11 30 20

13. Refer to Exhibit 2. Test that the mean difference between the two population means is 5. What is the value of test
statistic? What is the 𝒑-value for the hypothesis?
(A) 1.43~1.46; 0.05~0.10 (B) 1.43~1.46; 0.10~0.20 (C) 1.35~1.38; 0.20~0.40
(D) 1.35~0.38; 0.10~0.20 (E) 3.90~4.20; < 0.01
14. Refer to Exhibit 2. Provide a 90% confidence interval for the difference between the two population means.
(A) 4.178~10.822 (B) 3.958~10.042 (C) 4.965~10.035 (D) 4.985~10.015 (E) 4.140~10.860
Exhibit 3: (題組 15 ~17)
A paper company produces a 30 mils thick paper. In order to ensure that the thickness of the paper, a test is conducted
to see if the mean thickness is different from 30 mils. Assume the population standard deviation is 4 mils and a
sample of 256 papers had a mean thickness of 30.3 mils. Use 𝛼 = 0.05.
15. Refer to Exhibit 3. Compute the test statistics of this test. What is the 𝑝-value of this test?
(A) 1.1~1.3; 0.1151 (B) 0.2~0.4; 0.3821 (C) 1.1~1.3; 0.2302 (D) 0.2~0.4; 0.7624 (E)2.3~2.5; 0.0164
16. Refer to Exhibit 3. If 𝜇 = 29.8 mils is true, what is the probability of making Type II error?
(A) 0.1259 (B) 0.8741 (C) 0.9315 (D) 0.0685 (E) 0.9971
17. Refer to Exhibit 3. What is the power at 𝜇 = 30.3?
(A) 0.7756 (B) 0.7764 (C) 0.2236 (D) 0.1259 (E) 0.2244
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Exhibit 4: (題組 18 ~19)
Consider the following data for two independent random samples taken from two normal populations.
Sample 1 22 20 25 21 19 24 26 23 21 18 23
Sample 2 32 27 30 20 26 20 28 17 25

18. Refer to Exhibit 4. Compute the two sample variances. What is the degrees of freedom for the 𝓉 distribution?
(A) 5~6, 25~26; 11 (B) 5~6, 22~23; 12 (C) 6~7, 22~23; 11 (D) 6~7, 25~26; 11 (E) 6~7, 25~26; 12
19. Refer to Exhibit 4. What is the 95% confidence interval estimate of the difference between the two population
means?
(A) -7.2~-7; 1~1.2 (B) -7~-6.8; 1~1.2 (C) -6.7~-6.5; 0.5~0.7 (D) -6.4~-6.2; 0.2~0.4 (E) -8~-7.6; 1.8~2.2

Exhibit 5: (題組 20~21)

Consider the hypothesis test 𝐻0 : 𝜇 ≥ 25 and 𝐻𝑎 : 𝜇 < 25. The sample size is 100 and the population variance is 25.
Use 𝛼 = 0.01.
20. Refer to Exhibit 5. If the population mean is 24. What is the probability that the sample mean leads to the
conclusion do not reject 𝐻0 ?
(A) 0.9732 (B) 0.6293 (C) 0.0268 (D) 0.3707 (E) 0.9726
21. Refer to Exhibit 5. What is the power of the statistical test when the population mean is 24.5?
(A) 0.9834 (B) 0.9082 (C) 0.0166 (D) 0.0918 (E) 0.0934

Exhibit 6: (題組 22~23)

A large automobile insurance company selected samples of single and married male policy-holders and recorded the
number who made an insurance claim over the preceding three-year period. Suppose the survey, 185 of 500 single
policyholders recorded insurance claim and 319 of 1000 married policyholders recorded insurance claim.
22. Refer to Exhibit 6. What is the 90% confidence interval estimate of the difference between the two population
proportions?
(A) 0.0175~0.0845 (B) 0.0080~0.0940 (C) 0.0084~0.0936 (D) 0.0178~0.0842 (E) -0.0002~0.1022
23. Refer to Exhibit 6. Test to determine whether the claim rates differ between single and married male
policyholders. What is the value of the test statistic? What is the 𝒑-value for the hypothesis?
(A) 1.951; 0.0256 (B) 1.971; 0.0244 (C) 1.971; 0.0488 (D) 1.951; 0.0512 (E) 1.973; 0.0488

Exhibit 7: (題組 24~25)

Some people who bought XBox gaming systems complained about having received defective systems. The XBox
Company claims that the percentage of defective systems is less than 20%. A test is conducted to see if their claim is
achieved. In a sample of 120 units sold, 30 units were defective.
24. Refer to Exhibit 7. Compute the standard error of 𝑝. At 95% confidence using the critical value approach, what is
the critical value for this test?
(A) 0.0395; -1.645 (B) 0.0365; 1.645 (C) 0.0395; 1.645 (D) 0.0365; -1.645 (E) 0.0365; 1.96
25. Refer to Exhibit 7. What is the 𝑝-value of this test?
(A) 0.1038 (B) 0.1003 (C) 0.0869 (D) 0.1020 (E) 0.0853
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26. Consider the test 𝐻0 : 𝜇1 − 𝜇2 ≤ 0 and 𝐻𝑎 : 𝜇1 − 𝜇2 > 0. The following data for two independent random
samples taken from two normal populations. Sample 1: 𝒏𝟏 =10 𝒙𝟏 = 𝟐𝟐𝟓 𝒔𝟏 = 𝟒𝟎; Sample2: 𝒏𝟐 = 𝟏𝟐
𝒙𝟐 = 𝟏𝟖𝟔 𝒔𝟐 = 𝟑𝟎. Use 𝛼 = 0.05. What are critical value and conclusion?
(A) 1.746; Reject 𝐻0 (B) 1.740; Reject 𝐻0 (C) 1.725; Reject 𝐻0
(D) 1.746; Not reject 𝐻0 (E) 1.740; Not reject 𝐻0

27. Given the following information for the hypothesis testing of population mean, the sample size of 18, sample
mean of 51, and sample variance of 9: 𝐻0 : 𝜇 ≥ 52 𝐻𝑎 : 𝜇 < 52. What is the 𝒑-value of this test?
(A) 0.025~0.05 (B) 0.10~0.20 (C) > 0.20 (D) 0.05~0.10 (E) 0.20~0.40

28. The average expenditure on Valentine’s Day was expected to be $100.89. The average expenditure in a sample
survey of 50 male consumers was $135.67, and the average expenditure in a sample survey of 40 female
consumers was $99.64. Based on past surveys, the standard deviation for male consumers is assumed to be $45,
and the standard deviation for female consumers is assumed to be $35. Develop a 98% confidence interval for the
difference between the two population means.
(A) 19.50~52.56 (B) 16.414~55.646 (C) 14.305~57.755 (D) 22.157~49.903 (E) 25.218~46.842

29. The standard weight of a bottle of soft drink is 87 ounces with a standard deviation of the population 9 ounces. A
sample of 36 bottles with an average weight of 89.5 ounces is tested to determine whether or not the
manufacturing process is stable. What is the 𝒑-value of this test?
(A) 0.0122 (B) 0.0475 (C) 0.0950 (D) 0.0244 (E) 0.0930

30. A Businessweek/Harris survey asked senior executives at large corporations their opinions about the economic
outlook for the future. One question was, “Do you think that there will be an increase in the number of full-time
employees at your company over the next 12 months?” In the current survey, 9 of 132 executives answered Yes,
while in a previous year survey, 5 of 268 executives had answered Yes. Do these data suggest a statistically
significant difference between the proportions at the two points in time. Use a 0.02 level of significance. What is
the 𝒑-value, and what is your conclusion?
(A) 0.0114; significant difference (B) 0.0057; significant difference (C) 0.0348; not significant difference
(D) 0.0174; significant difference (E) 0.0366; not significant difference