SELF test STATISTICS FOR BUSINESS AND ECONOMICS

400統計學Statistics for Business and Economics

　習　題

方法

19. 考慮以下的假設檢定。

H0：μd ≤ 0

Ha：μd > 0

下列資料乃取自兩母體的配對樣本。

414 STATISTICS FOR BUSINESS AND ECONOMICS

The following data are from matched samples taken from two populations.

a. Compute the difference value for each element.b. Compute .c. Compute the standard deviation sd.d. Conduct a hypothesis test using α � .05. What is your conclusion?

22. The following data are from matched samples taken from two populations.

d̄

a. Compute the difference value for each element.b. Compute .c. Compute the standard deviation sd.d. What is the point estimate of the difference between the two population means?e. Provide a 95% confidence interval for the difference between the two population means.

Applications23. A market research firm used a sample of individuals to rate the purchase potential of a par-

ticular product before and after the individuals saw a new television commercial about theproduct. The purchase potential ratings were based on a 0 to 10 scale, with higher values in-dicating a higher purchase potential. The null hypothesis stated that the mean rating“after” would be less than or equal to the mean rating “before.” Rejection of this hypothesiswould show that the commercial improved the mean purchase potential rating. Use α � .05and the following data to test the hypothesis and comment on the value of the commercial.

d̄

testSELF

Population

Element 1 2

1 21 202 28 263 18 184 20 205 26 24

Population

Element 1 2

1 11 82 7 83 9 64 12 75 13 106 15 157 15 14

Purchase Rating Purchase Rating

Individual After Before Individual After Before

1 6 5 5 3 52 6 4 6 9 83 7 7 7 7 54 4 3 8 6 6

ASW_SBE_9e_Ch_10.qxd 10/31/03 1:54 PM Page 414

元素

母體

a. 請計算每一元素的差異值。

b. 請計算 d。

c. 請計算標準差 sd。

d. 以 α＝0.05 進行假設檢定，你的結論為何？

20. 下列資料乃取自兩母體的配對樣本。





d̄




d̄

testSELF

Population

Element 1 2

1 21 202 28 263 18 184 20 205 26 24

Population

Element 1 2

1 11 82 7 83 9 64 12 75 13 106 15 157 15 14



1 6 5 5 3 52 6 4 6 9 83 7 7 7 7 54 4 3 8 6 6


元素

母體

a. 請計算每一元素的差異值。

b. 請計算 d。

c. 請計算標準差 sd。

d. 兩母體平均數之差的點估計值為何？

e. 請計算兩母體平均數之差的 95% 信賴區間。

應用

21. 某家市場調查公司以一些民眾為樣本，評估他們在觀看某產品的新電視廣告前

後的購買可能性。此購買可能性評比的尺度為由 0 到 10，數字愈高表示購買

的可能性愈大。虛無假設為觀看「後」的平均評比會小於或等於觀看「前」的

平均評比，如拒絕此假設，即意味著這個新廣告確可提高購買可能性的平均評

比。請在 α＝0.05 之下，使用下列資料做假設檢定，並就此廣告的價值提出你

testSELF

testSELF

大統計學ch10.indd 400 2011/5/16 下午 10:22:24

401Chatper 10　兩母體平均數與比例的統計推論

的看法。





d̄




d̄

testSELF

Population

Element 1 2

1 21 202 28 263 18 184 20 205 26 24

Population

Element 1 2

1 11 82 7 83 9 64 12 75 13 106 15 157 15 14



1 6 5 5 3 52 6 4 6 9 83 7 7 7 7 54 4 3 8 6 6


民眾民眾之後之後之前之前

購買評比購買評比

22. 比較當季與上一季獲利的每股獲利資料在 CD 光碟中名為 Earnings2005 (The

Wall Street Journal, January 27, 2006)。請計算當季與上一季的母體平均數差異的

95% 信賴區間估計值。

23. 美國銀行的消費者支出調查蒐集每年信用卡支付的七項開支：交通、雜貨、

外出用餐、家庭開支、家居用品、服裝和娛樂 (U.S. Airways Attaché, December

2003)。使用一樣本大小為 42 個信用卡帳戶，假設每一帳戶均可用來確定每年

的信用卡支付雜貨 (母體 1) 和信用卡支付外出用餐 (母體 2) 的費用。使用差別

的資料，樣本平均差為 d＝$850，且樣本標準差為 sd＝$1123。

a. 建立虛無與對立假設以檢定每年信用卡支付雜貨和信用卡支付外出用餐的費

用沒有差異。

b. 使用 0.05 顯著水準，你可以確定兩母體平均數有差異嗎？p 值為何?

c. 雜貨和外出用餐中，哪一項開支的信用卡支付費用較高？兩母體平均數之差

的點估計值為何？母體平均數差異的 95% 信賴區間估計值為何？

24. 航空公司旅客往往根據飛行成本選擇機場搭機起飛。蒐集由俄亥俄州達頓市和

肯塔基州路易斯維爾市飛往 8 個城市的航班為一樣本的成本資料 (美元)，以幫

助確定兩個機場中哪個較為昂貴 (The Cincinnati Enquirer, February 19, 2006)。一

位研究者堅信由達頓市飛出較路易斯維爾市貴很多，請使用樣本資料確認是否

支持該研究者的論點。使用 α＝0.05 顯著水準。

25. In recent years, a growing array of entertainment options competes for consumer time. By2004, cable television and radio surpassed broadcast television, recorded music, and thedaily newspaper to become the two entertainment media with the greatest usage (The WallStreet Journal, January 26, 2004). Researchers used a sample of 15 individuals and col-lected data on the hours per week spent watching cable television and hours per week spentlistening to the radio.

10.3 Inferences About the Difference Between Two Population Means: Matched Samples 415

Individual Television Radio Individual Television Radio

1 22 25 9 21 212 8 10 10 23 233 25 29 11 14 154 22 19 12 14 185 12 13 13 14 176 26 28 14 16 157 22 23 15 24 238 19 21

fileCDTVRadio

Destination Dayton Louisville

Chicago-O’Hare $319 $142Grand Rapids, Michigan 192 213Portland, Oregon 503 317Atlanta 256 387Seattle 339 317South Bend, Indiana 379 167Miami 268 273Dallas–Ft. Worth 288 274

a. Use a .05 level of significance and test for a difference between the population meanusage for cable television and radio. What is the p-value?

b. What is the sample mean number of hours per week spent watching cable television?What is the sample mean number of hours per week spent listening to radio? Whichmedium has the greater usage?

26. StreetInsider.com reported 2002 earnings per share data for a sample of major companies(February 12, 2003). Prior to 2002, financial analysts predicted the 2002 earnings per sharefor these same companies (Barron’s, September 10, 2001). Use the following data to com-ment on differences between actual and estimated earnings per share.

Company Actual Predicted

AT&T 1.29 0.38American Express 2.01 2.31Citigroup 2.59 3.43Coca-Cola 1.60 1.78DuPont 1.84 2.18ExxonMobil 2.72 2.19General Electric 1.51 1.71Johnson & Johnson 2.28 2.18McDonald’s 0.77 1.55Wal-Mart 1.81 1.74

fileCDEarnings

71883_10_ch10_p393-433.qxd 12/27/09 3:00 PM Page 415

目的地達頓路易斯維爾

芝加哥－O’Hare 密西根州大湍城俄勒岡州波特蘭亞特蘭大西雅圖印第安納州南彎邁阿密達拉斯－沃爾斯堡

25. 近幾年中，日益增多的娛樂選項爭奪消費者時間。到 2004 年，有線電視及廣

播電台超過無線電視、音樂錄音與每日報紙，成為兩個最大使用量的娛樂媒體

(The Wall Street Journal, January 26, 2004)。研究人員使用 15 個人的樣本，蒐集

每週花在看有線電視與收聽廣播電台的時間數據。

fileCDAirFare

fileCDEarnings2005

大統計學ch10.indd 401 2011/5/16 下午 10:22:24

402統計學Statistics for Business and Economics

25. In recent years, a growing array of entertainment options competes for consumer time. By2004, cable television and radio surpassed broadcast television, recorded music, and thedaily newspaper to become the two entertainment media with the greatest usage (The WallStreet Journal, January 26, 2004). Researchers used a sample of 15 individuals and col-lected data on the hours per week spent watching cable television and hours per week spentlistening to the radio.

10.3 Inferences About the Difference Between Two Population Means: Matched Samples 415

Individual Television Radio Individual Television Radio

1 22 25 9 21 212 8 10 10 23 233 25 29 11 14 154 22 19 12 14 185 12 13 13 14 176 26 28 14 16 157 22 23 15 24 238 19 21

fileCDTVRadio

Destination Dayton Louisville

Chicago-O’Hare $319 $142Grand Rapids, Michigan 192 213Portland, Oregon 503 317Atlanta 256 387Seattle 339 317South Bend, Indiana 379 167Miami 268 273Dallas–Ft. Worth 288 274

a. Use a .05 level of significance and test for a difference between the population meanusage for cable television and radio. What is the p-value?

b. What is the sample mean number of hours per week spent watching cable television?What is the sample mean number of hours per week spent listening to radio? Whichmedium has the greater usage?



AT&T 1.29 0.38American Express 2.01 2.31Citigroup 2.59 3.43Coca-Cola 1.60 1.78DuPont 1.84 2.18ExxonMobil 2.72 2.19General Electric 1.51 1.71Johnson & Johnson 2.28 2.18McDonald’s 0.77 1.55Wal-Mart 1.81 1.74

fileCDEarnings

71883_10_ch10_p393-433.qxd 12/27/09 3:00 PM Page 415

個人個人有線電視有線電視廣播電台廣播電台

a. 使用 0.05 顯著水準檢定有線電視與廣播電台的母體平均使用量的差異。p 值

為何？

b. 每週看有線電視的樣本平均時間為何？每週收聽廣播電台的樣本平均時間為

何？哪一個媒體使用量較大？

26. StreetInsider.com 報導 2002 年主要公司之每股盈餘抽樣資料 (February, 12, 2003)。

在 2002 年以前，財務分析師已預測這些公司於 2002 年的每股盈餘 (Barron’s,

September 10, 2001)。請使用下列資料評論每股盈餘之實際與估計值的差異。


27. A survey was made of Book-of-the-Month-Club members to ascertain whether membersspend more time watching television than they do reading. Assume a sample of 15 re-spondents provided the following data on weekly hours of television watching and weeklyhours of reading. Using a .05 level of significance, can you conclude that Book-of-the-Month-Club members spend more hours per week watching television than reading?


a. Use α � .05 and test for any difference between the population mean actual and popu-lation mean estimated earnings per share.What is thep-value?What is your conclusion?

b. What is the point estimate of the difference between the two means? Did the analyststend to underestimate or overestimate the earnings?

c. At 95% confidence, what is the margin of error for the estimate in part (b)? Whatwould you recommend based on this information?

29. A manufacturer produces both a deluxe and a standard model of an automatic sander de-signed for home use. Selling prices obtained from a sample of retail outlets follow.

Respondent Television Reading Respondent Television Reading

1 10 6 9 4 72 14 16 10 8 83 16 8 11 16 54 18 10 12 5 105 15 10 13 8 36 14 8 14 19 107 10 14 15 11 68 12 14

fileCDTVRead


AT&T 1.29 0.38American Express 2.01 2.31Citigroup 2.59 3.43Coca-Cola 1.60 1.78DuPont 1.84 2.18Exxon-Mobil 2.72 2.19General Electric 1.51 1.71Johnson & Johnson 2.28 2.18McDonald’s 0.77 1.55Wal-Mart 1.81 1.74

fileCDEarnings

Model Price ($) Model Price ($)

Retail Outlet Deluxe Standard Retail Outlet Deluxe Standard

1 39 27 5 40 302 39 28 6 39 343 45 35 7 35 294 38 30


公司實際值預測值

a. 在 α＝0.05 情形下，請檢定每股盈餘之實際母體平均數與預測母體平均數間

是否存在差異。其 p 值為何？你的結論為何？

b. 此兩平均數之差的點估計值為何？究竟該分析師係低估或是高估了每股盈

餘？

c. 在 95% 信賴水準下，(b) 之邊際誤差為何？根據此資訊，你的建議為何？

27. 某製造商生產豪華與標準兩種款式的家用自動砂輪機，由各零售店樣本所得之

銷售價格如下。


27. A survey was made of Book-of-the-Month-Club members to ascertain whether membersspend more time watching television than they do reading. Assume a sample of 15 re-spondents provided the following data on weekly hours of television watching and weeklyhours of reading. Using a .05 level of significance, can you conclude that Book-of-the-Month-Club members spend more hours per week watching television than reading?


a. Use α � .05 and test for any difference between the population mean actual and popu-lation mean estimated earnings per share.What is thep-value?What is your conclusion?

b. What is the point estimate of the difference between the two means? Did the analyststend to underestimate or overestimate the earnings?

c. At 95% confidence, what is the margin of error for the estimate in part (b)? Whatwould you recommend based on this information?

29. A manufacturer produces both a deluxe and a standard model of an automatic sander de-signed for home use. Selling prices obtained from a sample of retail outlets follow.

Respondent Television Reading Respondent Television Reading

1 10 6 9 4 72 14 16 10 8 83 16 8 11 16 54 18 10 12 5 105 15 10 13 8 36 14 8 14 19 107 10 14 15 11 68 12 14

fileCDTVRead


AT&T 1.29 0.38American Express 2.01 2.31Citigroup 2.59 3.43Coca-Cola 1.60 1.78DuPont 1.84 2.18Exxon-Mobil 2.72 2.19General Electric 1.51 1.71Johnson & Johnson 2.28 2.18McDonald’s 0.77 1.55Wal-Mart 1.81 1.74

fileCDEarnings

Model Price ($) Model Price ($)

Retail Outlet Deluxe Standard Retail Outlet Deluxe Standard

1 39 27 5 40 302 39 28 6 39 343 45 35 7 35 294 38 30


零售店零售店豪華型豪華型標準型標準型

價格 ($) 價格 ($)

a. 該製造商對這兩種款式的建議零售價相差 $10。在 0.05 顯著水準下，請檢定

此兩款平均售價之差為 $10。

fileCDEarnings

fileCDTVRead

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403Chatper 10　兩母體平均數與比例的統計推論

b. 這兩種款式的平均售價之差的 95% 信賴區間為何？

10.4　兩母體比例之差的推論

令 p1 表母體 1 之比例，p2 表母體 2 的比例，我們再考量兩母體比例之差 p1－

p2 的推論。我們分別自母體 1 與母體 2 抽出 n1 與 n2 單位之兩組獨立隨機樣本。

p1－p2 之區間估計

在下述例題中，我們將介紹如何計算兩母體比例之差的邊際誤差，並建立其區

間估計。

某稅務代理公司想比較旗下兩家地區辦事處的工作品質，遂從各辦事處代理過

的退稅案件中，隨機抽取若干為一樣本，審查其正確性，以估計各辦事處退稅錯誤

案件所佔的比例。該公司特別想知道的是這兩個比例有否差異。令

p1＝母體 1 (辦事處 1) 的退稅錯誤比例

p2＝母體 2 (辦事處 2) 的退稅錯誤比例

p1＝母體 1 所抽取之簡單隨機樣本的樣本比例

p2＝母體 2 所抽取之簡單隨機樣本的樣本比例

因此，這兩個母體比例之間真正的差異為 p1－p2，其點估計量如下。

兩母體比例之差的點估計量

Chapter 10 Statistical Inference About Means and Proportions with Two Populations 417

a. The manufacturer’s suggested retail prices for the two models show a $10 price dif-ferential. Use a .05 level of significance and test that the mean difference between theprices of the two models is $10.

b. What is the 95% confidence interval for the difference between the mean prices of thetwo models?

10.4 Inferences About the Difference Between TwoPopulation ProportionsLetting p1 denote the proportion for population 1 and p2 denote the proportion for popula-tion 2, we next consider inferences about the difference between the two population pro-portions: p1 � p2. To make an inference about this difference, we will select twoindependent random samples consisting of n1 units from population 1 and n2 units frompopulation 2.

Interval Estimation of p1 � p2

In the following example, we show how to compute a margin of error and develop an in-terval estimate of the difference between two population proportions.

A tax preparation firm is interested in comparing the quality of work at two of its re-gional offices. By randomly selecting samples of tax returns prepared at each office andverifying the sample returns’ accuracy, the firm will be able to estimate the proportion oferroneous returns prepared at each office. Of particular interest is the difference betweenthese proportions.

The difference between the two population proportions is given by p1 � p2. The point esti-mator of p1 � p2 is as follows.

p1 �

p2 �

p̄1 �

p̄2 �

proportion of erroneous returns for population 1 (office 1)


sample proportion for a simple random sample from population 1


Thus, the point estimator of the difference between two population proportions is the dif-ference between the sample proportions of two independent simple random samples.

As with other point estimators, the point estimator 1 � 2 has a sampling distributionthat reflects the possible values of 1 � 2 if we repeatedly took two independent randomsamples. The mean of this sampling distribution is p1 � p2 and the standard error of 1 � 2

is as follows:

(10.11)σp̄1� p̄2� �p1(1 � p1)

n1�

p2(1 � p2)n2

p̄p̄p̄p̄

p̄p̄

POINT ESTIMATOR OF THE DIFFERENCE BETWEEN TWO POPULATIONPROPORTIONS

(10.10)p̄1 � p̄2


(10.10)　

亦即，兩母體間比例之差的點估計量，乃兩獨立簡單隨機樣本的樣本比例之差。

如同其他點估計量，若我們重複抽取兩個獨立隨機樣本，則點估計量 p1－p2 即

具有 p1－p2 各種可能數值的抽樣分配。此抽樣分配之平均值為 p1－p2，而 p1－p2

之標準誤則為：

p1－p2 之標準誤

Chapter 10 Statistical Inference About Means and Proportions with Two Populations 417

a. The manufacturer’s suggested retail prices for the two models show a $10 price dif-ferential. Use a .05 level of significance and test that the mean difference between theprices of the two models is $10.

b. What is the 95% confidence interval for the difference between the mean prices of thetwo models?

10.4 Inferences About the Difference Between TwoPopulation ProportionsLetting p1 denote the proportion for population 1 and p2 denote the proportion for popula-tion 2, we next consider inferences about the difference between the two population pro-portions: p1 � p2. To make an inference about this difference, we will select twoindependent random samples consisting of n1 units from population 1 and n2 units frompopulation 2.

Interval Estimation of p1 � p2

In the following example, we show how to compute a margin of error and develop an in-terval estimate of the difference between two population proportions.

A tax preparation firm is interested in comparing the quality of work at two of its re-gional offices. By randomly selecting samples of tax returns prepared at each office andverifying the sample returns’ accuracy, the firm will be able to estimate the proportion oferroneous returns prepared at each office. Of particular interest is the difference betweenthese proportions.

The difference between the two population proportions is given by p1 � p2. The point esti-mator of p1 � p2 is as follows.

p1 �

p2 �

p̄1 �

p̄2 �





Thus, the point estimator of the difference between two population proportions is the dif-ference between the sample proportions of two independent simple random samples.

As with other point estimators, the point estimator 1 � 2 has a sampling distributionthat reflects the possible values of 1 � 2 if we repeatedly took two independent randomsamples. The mean of this sampling distribution is p1 � p2 and the standard error of 1 � 2

is as follows:

(10.11)σp̄1� p̄2� �p1(1 � p1)

n1�

p2(1 � p2)n2

p̄p̄p̄p̄

p̄p̄

POINT ESTIMATOR OF THE DIFFERENCE BETWEEN TWO POPULATIONPROPORTIONS

(10.10)p̄1 � p̄2


(10.11)　

若樣本數目足夠大，且 n1 p1、n1 (1－p1)、n2 p2 與 n2 (1－p2) 均大於或等於 5，則

p1－p2 的抽樣分配將趨近於常態分配。

如同前述，其區間估計可以點估計值±邊際誤差表示。故兩母體比例之差的估

計可以下述形式表之：

p1－ p2 ±邊際誤差

因 p1－ p2 之抽樣分配趨近常態分配，我們可以


If the sample sizes are large enough that n1p1, n1(1 � p1), n2p2, and n2(1 � p2) are all greaterthan or equal to 5, the sampling distribution of 1 � 2 can be approximated by a normaldistribution.

As we showed previously, an interval estimate is given by a point estimate � a marginof error. In the estimation of the difference between two population proportions, an inter-val estimate will take the following form:

With the sampling distribution of 1 � 2 approximated by a normal distribution, wewould like to use zα/2 as the margin of error. However, given by equation(10.11) cannot be used directly because the two population proportions, p1 and p2, are un-known. Using the sample proportion 1 to estimate p1 and the sample proportion 2 to es-timate p2, the margin of error is as follows.

(10.12)

The general form of an interval estimate of the difference between two population propor-tions is as follows.

Margin of error � zα/2�p̄1(1 � p̄1)n1

�p̄2(1 � p̄2)

n 2

p̄p̄

σp̄1� p̄2σp̄1� p̄2

p̄p̄

p̄1 � p̄2 � Margin of error

p̄p̄

Returning to the tax preparation example, we find that independent simple random samples from the two offices provide the following information.

The sample proportions for the two offices follow.

The point estimate of the difference between the proportions of erroneous tax returns forthe two populations is 1 � 2 � .14 � .09 � .05. Thus, we estimate that office 1 has a .05,or 5%, greater error rate than office 2.

p̄p̄

p̄1 �35

250� 0.14

p̄2 �27

300� 0.09

fileCDTaxPrep

INTERVAL ESTIMATE OF THE DIFFERENCE BETWEEN TWO POPULATIONPROPORTIONS

(10.13)

where 1 � α is the confidence coefficient.

p̄1 � p̄2 � zα/2�p̄1(1 � p̄1)n1

�p̄2(1 � p̄2)

n 2

Office 1 Office 2n1 � 250 n2 � 300

Number of returns with errors � 35 Number of returns with errors � 27


代表邊際誤差。然而

大統計學ch10.indd 403 2011/5/16 下午 10:22:25

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SELF test STATISTICS FOR BUSINESS AND ECONOMICS