Statistics Midterm Test 107/2 (A 卷)

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[Notes]The attached probability value is the probability in upper tail of 𝔃 𝐨𝐫 𝓽 distribution
𝔃𝟎.𝟑 = 𝟎. 𝟓𝟐𝟒 𝔃𝟎.𝟐 = 𝟎. 𝟖𝟒𝟐 𝔃𝟎.𝟏𝟓 = 𝟏. 𝟎𝟑𝟔 𝔃𝟎.𝟏𝟎 = 𝟏. 𝟐𝟖𝟐 𝔃𝟎.𝟎𝟓 = 𝟏. 𝟔𝟒𝟓 𝔃𝟎.𝟎𝟐𝟓 = 𝟏. 𝟗𝟔 𝔃𝟎.𝟎𝟏 = 𝟐. 𝟑𝟐𝟔
𝔃𝟎.𝟎𝟎𝟓 = 𝟐. 𝟓𝟕𝟔 𝔃𝟎.𝟎𝟎𝟐𝟔 = 𝟐. 𝟖 𝓽𝟎.𝟎𝟐𝟓 (𝟓) = 𝟐. 𝟓𝟕𝟏 𝓽𝟎.𝟎𝟓 (𝟓) = 𝟐. 𝟎𝟏𝟓 𝓽𝟎.𝟎𝟐𝟓 (𝟔) = 𝟐. 𝟒𝟒𝟕 𝓽𝟎.𝟎𝟓 (𝟔) = 𝟏. 𝟗𝟒𝟑
𝓽𝟎.𝟎𝟏 (𝟏𝟏) = 𝟐. 𝟕𝟏𝟖 𝓽𝟎.𝟎𝟐𝟓 (𝟏𝟏) = 𝟐. 𝟐𝟎𝟏 𝓽𝟎.𝟎𝟓 (𝟏𝟏) = 𝟐. 𝟎𝟏𝟓 𝓽𝟎.𝟎𝟏 (𝟏𝟐) = 𝟐. 𝟔𝟖𝟏 𝓽𝟎.𝟎𝟐𝟓 (𝟏𝟐) = 𝟐. 𝟏𝟕𝟗
𝓽𝟎.𝟎𝟓 (𝟏𝟐) = 𝟏. 𝟕𝟖𝟐 𝓽𝟎.𝟎𝟐𝟓 (𝟐𝟐) = 𝟐. 𝟎𝟕𝟒 𝓽𝟎.𝟎𝟓 (𝟐𝟐) = 𝟏. 𝟕𝟏𝟕 𝓽𝟎.𝟎𝟎𝟓 (𝟑𝟕) = 𝟐. 𝟕𝟏𝟓 𝓽𝟎.𝟎𝟏 (𝟑𝟕) = 𝟐. 𝟒𝟑𝟏
𝓽𝟎.𝟎𝟎𝟓 (𝟑𝟖) = 𝟐. 𝟕𝟏𝟐 𝓽𝟎.𝟎𝟏 (𝟑𝟖) = 𝟐. 𝟒𝟐𝟗 𝓽𝟎.𝟎𝟎𝟓 (𝟗𝟗) = 𝟐. 𝟔𝟐𝟔 𝓽𝟎.𝟎𝟏 (𝟗𝟗) = 𝟐. 𝟑𝟔𝟒 𝓽𝟎.𝟎𝟐𝟓 (𝟗𝟗) = 𝟏. 𝟗𝟖𝟒
𝓽𝟎.𝟎𝟓 (𝟗𝟗) = 𝟏. 𝟔𝟔𝟎 𝓽𝟎.𝟏 (𝟗𝟗) = 𝟏. 𝟐𝟗𝟎 𝓽𝟎.𝟐 (𝟗𝟗) = 𝟎. 𝟖𝟒𝟓

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

1. In conducting a hypothesis test about 𝑝1 – 𝑝2 , any of the following approaches can be used except
(A) comparing the hypothesized difference to the confidence interval (B) comparing the 𝑝-value to 𝛼
(C) comparing the observed frequencies to the expected frequencies
(D) comparing the test statistic to the critical value
2. If the hypothesis test you are conducting is a two-tailed test, which of the following is a possible step that you
could take to increase the power of the test?
(A) Use the t-distribution (B) Increase beta (C) Reduce the sample size (D) Increase alpha
3. When testing the difference between two dependent population means, the test statistic is based on a
(A) Sum of the population variances (B) Pooled variance
(C) Standard deviation of the differences (D) Pooled proportion
4. The level of significance is the
(A) same as the confidence coefficient (B) maximum allowable probability of Type I error
(C) same as the 𝑝-value (D) maximum allowable probability of Type II error
5. If we are interested in testing whether the proportion of items in population 1 is larger than the proportion of
items in population 2, the
(A) null hypothesis should state 𝑝1 − 𝑝2 < 0 (B) alternative hypothesis should state 𝑝1 − 𝑝2 < 0
(C) null hypothesis should state 𝑝1 − 𝑝2 > 0 (D) alternative hypothesis should state 𝑝1 − 𝑝2 > 0
6. More evidence against 𝐻0 is indicated by
(A) smaller p-values (B) lower probabilities of a Type II error
(C) lower levels of significance (D) smaller critical values
7. If two independent large samples are taken from two populations, the sampling distribution of the difference
between the two sample means
(A) can be approximated by a normal distribution (B) will have a mean of one
(C) can be approximated by a Poisson distribution (D) will have a variance of one
8. What can be concluded by failing to reject the null hypothesis?
(A) that the alternative hypothesis is correct, therefore the null hypothesis should be rejected
(B) that the function of the alternative hypothesis is incorrect
(C) that the test procedure is not strong enough to reject the null hypothesis
(D) that the null hypothesis is correct, therefore the alternative hypothesis should be rejected
9. When estimating a confidence interval for the difference between 2 means using the method where sample
variances are pooled, which of the following assumptions is not needed?
 統計學(二) 0075 統計學 (A 卷) 本試題共 4 頁第 2 頁
(A) the populations have equal variance (B) the sample sizes are equal
(C) the populations are normally distributed (D) the samples are independent
10. Which of the following does not need to be known in order to compute the 𝑝-value?
(A) the level of significance (B) knowledge of whether the test is one-tailed or two-tailed
(C) the value of the test statistic (D) All of these are needed

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

Exhibit 1: (題組 11 – 12)
Periodically, customers are asked to evaluate the service of financial consultants. Higher ratings on the client
satisfaction survey indicate better service, with 70 the maximum service rating. Independent samples of service
ratings for two financial consultants are summarized here. Consultant A has 10 years of experience, whereas
consultant B has 1 year of experience. Use 𝛼 = 0.01 and test to see whether the consultant with more experience
has the higher population mean service rating.
Consultant A: 𝑛1 = 21, 𝑥1 = 51.48, 𝑠1 = 11.01; Consultant B: 𝑛2 = 23, 𝑥2 = 40.52, 𝑠2 = 17.15
11. What are the null and alternative hypothesis? What is the degrees of freedom for the t distribution?
(A) 𝐻0 : 𝜇1 = 𝜇2 vs. 𝐻𝑎 : 𝜇1 ≠ 𝜇2 ; 37 (B) 𝐻0 : 𝜇1 = 𝜇2 vs. 𝐻𝑎 : 𝜇1 ≠ 𝜇2 ; 38
(C) 𝐻0 : 𝜇1 ≥ 𝜇2 vs. 𝐻𝑎 : 𝜇1 < 𝜇2 ; 38 (D) 𝐻0 : 𝜇1 ≤ 𝜇2 vs. 𝐻𝑎 : 𝜇1 > 𝜇2; 38
(E) 𝐻0 : 𝜇1 ≤ 𝜇2 vs. 𝐻𝑎 : 𝜇1 > 𝜇2; 37
12. What are the test statistic, critical value and your conclusion?
(A) 2.544; 2.429; reject 𝐻0 (B) 2.544; 2.715; not reject 𝐻0 (C) 9.726; 2.429; reject 𝐻0
(D) 2.544; 2.431; reject 𝐻0 (E) 2.544; 2.712; not reject 𝐻0
Exhibit 2: (題組 13 - 16)
Annual expenditure for prescription drugs was $800 per person in Taipei. We are conducting a hypothesis test to
determine whether the sample data support the conclusion that the population annual expenditure for prescription
drugs per person is lower in Taichung than in Taipei. A sample of 64 individuals in Taichung showed a sample mean
of $710. Use a population standard deviation of $300 to answer the following questions. 𝛼 is set to be 0.01.
13. What is the value of the test statistic? What is the rejection rule?
(A) -2.4; -2.576 (B) -2.4; 2.326 (C) -2.4; -2.326 (D) -0.375; -2.576 (E) -0.375; -2.326
14. Which is the best conclusion for the test?
(A) Do not reject 𝐻0 . The annual expenditure in Taichung is significantly higher.
(B) Do not reject 𝐻0 . The annual expenditure in Taichung is insignificantly lower.
(C) Do not reject 𝐻0 . The annual expenditure in Taichung is insignificantly higher.
(D) Reject 𝐻0 . The annual expenditure in Taichung is significantly lower.
(E) Reject 𝐻0 . The annual expenditure in Taichung is significantly higher.
15. What is the power when the actual population annual expenditure in Taichung is 750?
(A) 0 ~ 0.15 (B) 0.15 ~ 0.2 (C) 0.2 ~ 0.3 (D) 0.3 ~ 0.7 (E) 0.7 ~ 0.85
16. Another research institute also conducted a survey for the hypothesis test 𝐻0 : 𝜇 = 800 v.s. 𝐻𝑎 : 𝜇 ≠ 800.
Based on their survey, [750, 815] is the 99% confidence interval estimate of 𝜇. According to the interval
estimate, which is a correct conclusion for the hypothesis test? Use 𝛼 = 0.01.
(A) Reject 𝐻0 , since (750+815)/2 is different from 710 (B) Reject 𝐻0 , since (750+815)/2 is larger than 710
(C) Reject 𝐻0 , since 710 is not in the interval [750, 815]
(D) Do not reject 𝐻0 , since (750+815)/2 is smaller than 800
(E) Do not reject 𝐻0 , since 800 is in the interval [750, 815]
 統計學(二) 0075 統計學 (A 卷) 本試題共 4 頁第 3 頁
Exhibit 3: (題組 17 - 18)
The graduate school entrance exam consists of two parts: Statistics and Calculus. Sample data showing the Statistics
and Calculus scores for a sample of 12 students who took the exam follow.
Student 1 2 3 4 5 6 7 8 9 10 11 12
Statistics 54 43 53 57 44 50 48 49 61 57 39 59
Calculus 47 38 46 61 42 52 43 46 61 54 33 61
17. What is the point estimate of the difference between the mean scores for the two tests? What is the degree of
freedom for the t distribution adopted?
(A) 2.5; 11 (B) 2.5; 22 (C) 2.5; 22 (D) 30; 11 (E) 30; 12
18. Use a 0.05 level of significance and test for a difference between the population mean for the Statistics scores
and the population mean for the Calculus scores. What are the test statistic and 𝑝-value?
(A) 2.209; 0.02~0.05 (B) 2.307; 0.01~0.025 (C) 2.307; 0.02~0.05
(D) 2.410; 0.01~0.025 (E) 2.410; 0.02~0.05
Exhibit 4: (題組 19 - 20)
A census showed that bachelor’s degree recipients with majors in business received average starting salaries of
$40,000 in 2012. The results for a sample of 100 business majors receiving a bachelor’s degree in 2013 showed a
mean starting salary of $41,000 with a sample standard deviation of $5000. Conduct a hypothesis test to determine
whether the mean starting salary for business majors in 2013 is greater than that in 2012.
19. What are the null hypothesis? What is the value of the test statistic?
(A) 𝐻0 : 𝜇 < 40000; 0.02 (B) 𝐻0 : 𝜇 ≤ 40000; -0.02 (C) 𝐻0 : 𝜇 ≥ 40000; -2
(D) 𝐻0 : 𝜇 ≤ 40000; 2 (E) 𝐻0 : 𝜇 = 40000; 2
20. What is the 𝑝-value?
(A) 0 ~ 0.01 (B) 0.01 ~ 0.025 (C) 0.02 ~ 0.05 (D) 0 ~ 0.02 (E) 0.05 ~ 0.1
Exhibit 5: (題組 21 - 22)
A company is redesigning the checkout lanes in its supermarkets throughout the country and is considering two
designs. Tests on customer checkout times conducted at a store where the two new systems have been installed
result in the following summary of the data.
System 1: 𝑛1 = 120, sample mean= 4.1(minutes), population standard deviation= 2.191(minutes)
System 2: 𝑛2 = 100, sample mean= 3.4(minutes), population standard deivation= 1.50(minutes)
21. Test at the 0.1 level of significance to determine whether the population mean checkout times of the two
systems differ. What is the 𝑝-value and your conclusion?
(A) 0.0026; two systems are significant difference (B) 0.0026; two systems are not significant difference
(C) 0.0052; two systems are significant difference (D) 0.0052; two systems are not significant difference
(E) 0.9974; two systems are not significant difference
22. What is the 90% confidence interval for the difference between two population mean (𝜇1 − 𝜇2 )? At 𝛼 = 0.10,
what is your conclusion?
(A) [0.29, 1.11]; 𝜇1 &𝜇2 are significant difference (B) [0.29, 1.11]; 𝜇1 &𝜇2 are not significant difference
(C) [0.21, 1.19]; 𝜇1 &𝜇2 are significant difference (D) [0.21, 1.19]; 𝜇1 &𝜇2 are not significant difference
(E) [-0.05, 1.45]; 𝜇1 &𝜇2 are not significant difference
Exhibit 6: (題組 23 - 24)
According to the research center of a chain department store, 5% of all merchandise sold in all their branches in
Taiwan gets returned. To test whether proportion of returns at the Taichung store is significantly different from the
returns in all branches in Taiwan. The branch in Taichung sampled 100 items sold in January and found that 1 of the
items were returned.
 統計學(二) 0075 統計學 (A 卷) 本試題共 4 頁第 4 頁
23. What is the 𝑝-value?
(A) 0 ~ 0.01 (B) 0.01 ~ 0.025 (C) 0.025 ~ 0.05 (D) 0.05 ~ 0.1 (E) 0.1 ~ 0.2
24. Adam uses a 0.1 level of significance, and Kelly uses a 0.01 level of significance. What will they conclude,
respectively, for their hypothesis testing?
(A) both “reject 𝐻0 ” (B) both “do not reject 𝐻0 ” (C) Adam: reject 𝐻0 ; Kelly: do not reject 𝐻0
(D) Adam: do not reject 𝐻0 ; Kelly: reject 𝐻0 (E) no conclusion
Exhibit 7: (題組 25 - 26)
Medical tests were conducted to learn about drug-resistant tuberculosis. Of 140 cases tested in Taichung, 14 were
found to be drug-resistant. Of 260 cases tested in Taipei, 13 were found to be drug-resistant. Do these data suggest a
statistically significant difference between the proportions of drug-resistant cases in the two cities?
25. Use the significance level 𝛼 = 0.05. What are the test statistic, critical value(s) and conclusion?
(A) 1.901; ±1.645; significant difference (B) 1.901; ±1.96; not significant difference
(C) 1.740; ±1.645; significant difference (D) 1.740; ±1.96; not significant difference
(E) 1.901; 1.645; significant difference
26. In order to construct the 95% confidence interval for the proportions difference of drug-resistant cases between
two cities, what is the margin of error?
(A) 0.0433 (B) 0.0473 (C) 0.05 (D) 0.0515 (E) 0.0563

27. Consider the following hypothesis test: 𝐻0 : 𝜇 = 18 v.s. 𝐻𝑎 : 𝜇 ≠ 18. A sample of 100 provided a sample
mean 17.5 and a sample standard deviation 2.8. Using a 0.05 level of significance, what is the critical value?
What is the conclusion for the hypothesis test?
(A) ±1.984; do not reject the null hypothesis (B) ±1.984; reject the null hypothesis
(C) ±1.660; do not reject the null hypothesis (D) ±1.660; reject the null hypothesis
(E) ±1.290; reject the null hypothesis
28. Consider the following hypothesis test: 𝐻0 : 𝜇 ≤ 15 v.s. 𝐻𝑎 : 𝜇 > 15. The population standard deviation is 8.
The sample size is 100. Using a 0.05 level of significance. What is the probability of making a type II error
when the actual population mean is 18?
(A) 0.95 ~ 0.975 (B) 0.975 ~ 0.99 (C) 0.01 ~ 0.05 (D) 0.025 ~ 0.05 (E) 0.01 ~ 0.025
29. The following results come from a paired-difference experiment with 6 pairs. These difference (treatment 1−
treatment 2) are -1, 2, 1, -1, 3, and -1. Find a 95% confidence interval for the difference between the two treatment
means. Does the confidence interval indicate a significant difference between the two treatment means?
(A) -1.259~2.259; not significant difference (B) -1.348~2.348; not significant difference
(C) -1.348~2.348; significant difference (D) -0.948~1.948; significant difference
(E) -0.948~1.948; not significant difference
30. Ten years ago, 50.45% of American families owned stocks. This year, a research institute sampled 121
American families and 52 of the 121 families said that they own stocks. Which is the smallest level of
significance for which you could draw the conclusion that the proportion of American families own stocks this
year is different from 0.5045? (Note: round off the value of the test statistic to the 3rd decimal place)
(A) 0.1 (B) 0.2 (C) 0.05 (D) 0.025 (E) 0.01