東吳大學104 學年度第二學期商用統計學期中考試卷4/20

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1. 請注意考試規則!

2. 直接在此作答，姓名: __________施楸容____________，學號:_________03154172_________

3. 另存新檔為學號

4. 記得儲存，作答完後，附件請寄至[email protected] 與[email protected] 5. 請記得cc或副本給自己

6. 從Excel的結果截圖重要的資訊複製貼上Word。

7. 回答問題時請詳加說明您的虛無假設與對立假設，以及樣本（母體）的定義為何（差異檢定時），根據您截圖哪裡做出什麼結論！

8. 每題請說明用到的公式是課本哪一頁哪一個公式代號。

題目開始

A. 根據所附的Excel 資料檔，回答以下問題：

1. 這是一個從東吳大學隨機抽樣的68個學生所回答的問題，CC數是他購買搖搖杯的果汁飲料CC數。搖搖杯的老闆宣稱她的飲料至少有435CC，如果您是學校的學生代表，站在學生的立場您會如何進行這樣宣稱的檢定？顯著水準為0.05。

P: 搖搖杯的果汁飲料CC數 H0:P 435

H¹: P<435

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p-Value < ，有足夠的證據拒絕H0，表示搖搖杯的果汁飲料CC數不會至少

t Test for Hypothesis of the Mean

Null Hypothesis m= 435

Level of Significance 0.05

Sample Size 68

Sample Mean 432.6323529

Sample Standard Deviation 6.130190549

Standard Error of the Mean 0.7434

Degrees of Freedom 67

t Test Statistic -3.1849

Lower-Tail Test Calculations Area

Lower Critical Value -1.6679 For one-tailed tests:

p-Value 0.0011 TDIST value 0.0010992

Reject the null hypothesis 1-TDIST value 0.9989008 Data

Intermediate Calculations

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435cc。

2. 同1，若以信賴區間confidence interval來估計，其上下界為何?

2

Data

Sample Standard Deviation 6.1301905

Sample Mean 432.63235

Sample Size 68

Confidence Level 95%

t Value 1.9960

Interval Half Width 1.4838

Confidence Interval

Interval Lower Limit 431.15

Interval Upper Limit 434.12

在 95%的信心水准下，搖搖杯飲料容量的信賴區間在（431.15 至 434.12）间。

3. 根據有無I Phone 檢定東吳大學的學生持有愛瘋的比例為0.5，顯著水準為

0.05。

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=37/68=0.544 Ho: =0.5 H1: 0.5

p-value< ，沒有足够的证据拒绝Ho，因此可以說東吳大學學生持有Iphone

的比例等於0.5。

4. 同3，東吳大學學生持有愛瘋手機比例的信賴區間confidence interval為何?

Z Test of Hypothesis for the Proportion

Null Hypothesis p = 0.5

Number of Items of Interest 37

Sample Size 68

Sample Proportion 0.544117647

Standard Error 0.0606

Z Test Statistic 0.7276

Two-Tail Test

Lower Critical Value -1.9600

Upper Critical Value 1.9600

p-Value 0.4669

Do not reject the null hypothesis Data

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在 95%信心水准下，檢定東吳大學的學生持有愛瘋的比例的信賴區間落在

（42.57 至 66.25%）間。

5. 請檢定東吳大學男學生持有愛瘋的比例是否顯著低於女生，顯著水準為0.5?

np^≧10,n(1-p^)≧10,近似常态分配

：20/68=0.294 ：48/68=0.706 Ho：p=0.5

H1:p≠0.5

P: 東吳大學男學生持有愛瘋的比例低於女生 Confidence Interval Estimate for the Proportion

Data

Sample Size 68

Number of Successes 37

Confidence Level 95%

Sample Proportion 0.544117647

Z Value -1.9600

Standard Error of the Proportion 0.0604

Interval Lower Limit 0.4257

Interval Upper Limit 0.6625

Confidence Interval

have Iphone No Iphone

boy 20 17 37

girl 17 14 31

37 31

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p-value<a/2，有足够的证据拒绝Ho，表示東吳大學男學生持有愛瘋的比例低於女生。

6. 同5，其東吳大學男女生持有愛瘋比例差距的信賴區間confidence interval為

何?

在 95%的信心水准下，東吳大學男女生持有愛瘋比例差距的信賴區間为

（-0.0896 至 0.0739 人）。

7. 如果我想檢定男生與女生所喝的搖搖杯的容量是否一樣，在顯著水準為0.05 下，請用信賴區間confidence interval的方法來說明是否相同。必要時請說明

Z Test for Differences in Two Proportions

Hypothesized Difference 0

Group 1

Number of Items of Interest 20 Confidence Level 50%

Sample Size 37

Group 2

Number of Items of Interest 17 Z Value -0.6745

Sample Size 31 Std. Error of the Diff. between two Proportions 0.1212

Group 1 Proportion 0.540540541

Group 2 Proportion 0.548387097 Interval Lower Limit -0.0896

Difference in Two Proportions -0.00784656 Interval Upper Limit 0.0739

Average Proportion 0.5441

Z Test Statistic -0.0647

Upper-Tail Test

p-Value 0.5258

Do not reject the null hypothesis

Data Confidence Interval Estimate

of the Difference Between Two Proportions Data

Z Test for Differences in Two Proportions

Group 1

Number of Items of Interest 20 Confidence Level 50%

Sample Size 37

Group 2

Number of Items of Interest 17 Z Value -0.6745

Sample Size 31 Std. Error of the Diff. between two Proportions 0.1212

Group 1 Proportion 0.540540541

Group 2 Proportion 0.548387097 Interval Lower Limit -0.0896

Difference in Two Proportions -0.00784656 Interval Upper Limit 0.0739

Z Test Statistic -0.0647

Upper-Tail Test

p-Value 0.5258

Do not reject the null hypothesis

of the Difference Between Two Proportions Data

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您選擇的公式理由。

：女生喝的飲料cc數

：男生喝的飲料cc數

： =

：

p-value< ，沒有足够的证据拒绝Ho，因此可以說信賴區間內也包含了0 代

F Test for Differences in Two Variances

Larger-Variance Sample

Sample Size 31

Sample Variance 40.44516129

Smaller-Variance Sample

Sample Size 37

F Test Statistic 1.1162

Population 1 Sample Degrees of Freedom 30 Population 2 Sample Degrees of Freedom 36

Calculations Area

Two-Tail Test FDIST value 0.3733

p-Value 0.7466

Pooled-Variance t Test for the Difference Between Two Means (assumes equal population variances)

Population 1 Sample

Sample Size 31 Confidence Level 95%

Sample Mean 432.6129

Population 2 Sample Degrees of Freedom 66

Sample Size 37 t Value 1.9966

Sample Mean 432.64865 Interval Half Width 3.0026

Interval Lower Limit -3.0383 Population 1 Sample Degrees of Freedom 30 Interval Upper Limit 2.9668 Population 2 Sample Degrees of Freedom 36

Total Degrees of Freedom 66

Pooled Variance 38.1483

Standard Error 1.5039

Difference in Sample Means -0.0357

t Test Statistic -0.0238

Two-Tail Test

Lower Critical Value -1.9966

p-Value 0.9811

Do not reject the null hypothesis Intermediate Calculations

for the Difference Between Two Means Data

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表無顯著差異表示男女生喝的飲料容量相同。

B. In the 2008 General Social Survey, participants were asked what their news source is. Among those surveyed in the 18-to 29-year-old age group, 92/262=.351 said that the Internet was their main news source. In the 50-and-over age group , only 59/632=.093 said that the Internet was their main news source. Use this sample information to determine a 99% confidence interval that estimate the parameter p1-p2 =difference between population proportion who use the Internet as their main source in the two age groups.

：18-to 29-year-old age group

：50-and-over age group H0:P1=P2

H1: P1>P2

1≥10 ,

¹

(1−

¹

)≥10 ,

²

≥10 ,

²

(1−

²

)≥10 ，近似常態分配

B

Data

Group 1

Sample Size 262

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Sample Size 632

Group 1 Proportion 0.351145038 Group 2 Proportion 0.09335443 Difference in Two Proportions 0.257790608

Z Test Statistic 9.3640

Upper-Tail Test

p-Value 0.0000

Reject the null hypothesis

Confidence Interval Estimate of the Difference Between Two Proportions

Data

Confidence Level 99%

Z Value -2.5758

Std. Error of the Diff. between two

Proportions 0.0317

Interval Lower Limit 0.1762

Interval Upper Limit 0.3394

p-Value > ，有足夠的證據拒絕H0，表示18至29歲年齡組使用互聯網作

為新聞來源多於50歲及以上年齡組。

C. B. A random sample of five college women was asked for their own heights and their mothers’ heights. The researchers wanted to know whether college women are taller on average than their mothers. The results (inches) follow:

Pair 1 2 3 4 5

Daughter 66 64 64 69 66

Mother 61 62 65 66 62

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Find the 95% confidence interval for the parameter of interest in this situation.

未知，樣本數很小，相依樣本採t分配

C

Data

Sample Standard Deviation 2.3021729

Sample Mean 2.6

Sample Size 5

Confidence Level 95%

t_Value 2.7764

Confidence Interval

Interval Lower Limit -0.26

Interval Upper Limit 5.46

在95%的信心水準下，女兒較母親身高高的信賴區間為-0.26到5.46

D. Following are data for 20 individuals on resting pulse rate and whether or not the individual regularly exercise.

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Person 1 2 3 4 5 6 7 8 9 10 Pulse 72 62 72 84 60 63 66 72 75 64 Regularly Exercise N Y Y N Y Y N N Y Y

Person 11 12 13 14 15 16 17 18 19 20 Pulse 62 84 76 60 52 60 64 80 60 64 Regularly Exercise N N N Y Y N Y Y N Y

Calculate a 95% confidence interval for the difference in means for the two groups.

0： 1= 2 1： 1≠ 2

1− 2：difference in variance resting pulse rate in a population that does not regularly exercise(group 1)

1：Yes 2：No

p-Value > α/2，沒有足夠的證據拒絕 0，表示有規律運動的Pluse變異數和沒有規律運動的 Pluse 變異數相等。

估計 1- 2的信賴區間， 1未知， 2未知，且 1= 2，採t分配(pooled) 0： 1= 2 1：

1≠ 2

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p-Value >α2，沒有足夠的證據拒絕 0，表示有規律運動的Pulse平均數和沒有規律運動的 Pulse平均數相等。

95%信心水準下，信賴區間為[-1.1652 13.6650]，區間包含零，表示有規律運動的Pulse平均數和沒有規律運動的Pulse平均數相等。

8. Extra point. 請檢定上題7男女生的手搖杯CC數變異數是否相等，顯著水準

為0.05。

：女生

：男生

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F檢定值趨近於1 可判斷兩者母體變異數相同

F Test for Differences in Two Variances

Larger-Variance Sample

Sample Size 31

Smaller-Variance Sample

Sample Size 37

F Test Statistic 1.1162

Population 1 Sample Degrees of Freedom 30 Population 2 Sample Degrees of Freedom 36

Calculations Area

Two-Tail Test F.DIST.RT value 0.3733

p-Value 0.7466