東吳大學104 學年度第二學期商用統計學期中考試卷

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1. 請注意考試規則!

2. 直接在此作答，姓名:张曼琴，

學號:03154174 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。

According textbook P.517 the steps for testing a single mean with a one sample t- test

虛無假設:店家飲料至少有435CC 對立假設:店家飲料不到435CC

P-value < level of significance拒絕虛無假設表示店家飲料不到435CC

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

According textbook P.429 calculating & interpreting a confidence interval for a population mean.

飲料容量的95%信賴區間為431.15~434.12

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。

According textbook P.480 the steps for a z-test for one proportion 虛無假設:東吳大學學生持有愛瘋比例為0.5

對立假設:東吳大學學生持有愛瘋比例不是0.5 Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation 6.1301905

Sample Mean 432.63235

Sample Size 68

Confidence Level 95%

Standard Error of the Mean 0.743394798

t Value 1.9960

Interval Half Width 1.4838

Interval Lower Limit 431.15 Interval Upper Limit 434.12

Confidence Interval

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3

P > level of significance無法拒絕虛無假設

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?

According textbook P.392 formula for a confidence interval for a population proportion p

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

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

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東吳大學學生持有愛瘋比例的信賴區間為0.4257~0.6625

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?

According P.490 the steps for a z-test for the difference in two proportions

虛無假設: 東吳大學男生持有Iphone比例低於女生對立假設: 東吳大學男生持有Iphone比例不低於女生 Let Group 1 be boy Group 2 be girl

P-value > level of significance無法拒絕虛無假設但信賴區間包含0 無顯著差異

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?

According textbook P.396 confidence interval for the difference in two population proportions

have Iphone No Iphone

boy 20 17 37

girl 17 14 31

37 31

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

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東吳大學男女生持有愛瘋比例差距的95%信賴區間為-0.0896~0.0739 無顯著差異

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。必要時請說明您選擇的公式理由。

檢定男女生母體的變異數是否相同利用F檢定 p1為女生p2為男生

虛無假設:男女生喝的飲料容量一樣對立假設: 男女生喝的飲料容量不一樣

Group 1

Sample Size 37

Group 2

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

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

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

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P-value > level of significance無法拒絕虛無假設信賴區間內也包含了0 代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.

According textbook P. 396 confidence interval for the difference in two population proportions and P.490 the steps for a z-test for the difference in two proportions 虛無假設:使用網路為主要訊息來源的人18-29歲與50歲以上比例相同對立假設: 使用網路為主要訊息來源的人18-29歲與50歲以上比例不相同 Let group 1 be the age 18-to 29-year-old age group

Group 2 be the 50-and –over age group

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

Difference in Sample Means -0.0357

t Test Statistic -0.0238

Two-Tail Test

p-Value 0.9811

Do not reject the null hypothesis Intermediate Calculations

for the Difference Between Two Means Data

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使用網路為主要訊息來源的人18-29歲與50歲以上比例差異的99信賴區間為 0.1762~0.3394 有顯著差異

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

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

Accordind textbook P. 521 the steps for a paired t-test

Group 1

Sample Size 262

Group 2

Sample Size 632 Std. Error of the Diff. between two Proportions 0.0317

Group 2 Proportion 0.09335443 Interval Lower Limit 0.1762

Difference in Two Proportions 0.257790608 Interval Upper Limit 0.3394

Z Test Statistic 9.3640

Two-Tail Test

p-Value 0.0000

Reject the null hypothesis

Pair 1 2 3 4 5

Daughter 66 64 64 69 66

Mother 61 62 65 66 62

Daughter-Mother 5 2 -1 3 4

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女兒與母親的平均身高差異95信賴區間為 2.6-2.78*1.0296 到 2.6+2.78*1.0296 之間 -0.262288~50462288之間

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

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.

首先以F檢定有規律運動者和無規律運動者兩母體變異數是否相等。

According textbook P.440 pooled confidence interval for the difference between two population means

P1為無規律運動者 P2為有規律運動者

Paired t Test

Hypothesized Mean Difference 0

Level of significance 0.05

Sample Size 5

DBar 2.6000

S_D 2.3022

t Test Statistic 2.5253

Lower Critical Value -2.1318 T.DIST.RT 0.0325

p-Value 0.9675 1-T.DIST.RT 0.9675

TDIST Calculations Lower-Tail Test

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95%信賴區間為-2.5135～13.6650之間

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。

Let P1 be girl, P2 be boy

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F檢定值趨近於1 可判斷兩者母體變異數相同

檢討（真心誠意）T.T

老師寫這篇檢討之前，我要向你和助教道歉，不好意思拿了這麼低分我都沒臉見你們了(哭)。但希望你們不要對我失望，因為我也快絕望了!!!! 現在我來檢討我自己。老師我真的有在努力學統計啊，考試前還很不好意思的一直助教，重點是老師你真的有一個超好超級好的助教，他真的很認真很有耐心的教導我這個統計智障。我也從對pooled 雙尾這些沒什麼概念，到開始會實際操作。但我真的不知道我為什麼當天考試突然就全部都不會了，可能前天根本沒睡頭腦當機，也許是我一直找不到拼音輸入法慌張了，也許是電腦換了幾台時間不夠…… 好老師我想知道這些都是藉口，我真的會放更多心思在統計學，多努力做練習，用excel的！

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