Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)
Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)
6th Edition
ISBN: 9781337115186
Author: David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, Jeffrey D. Camm, James J. Cochran
Publisher: Cengage Learning
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Chapter 12.2, Problem 12E

Vehicle Quality Ratings. A J. D. Power and Associates vehicle quality survey asked new owners a variety of questions about their recently purchased automobile. One question asked for the owner’s rating of the vehicle using categorical responses of average, outstanding, and exceptional. Another question asked for the owner’s education level with the categorical responses some high school, high school graduate, some college, and college graduate. Assume the sample data below are for 500 owners who had recently purchased an automobile.

Chapter 12.2, Problem 12E, Vehicle Quality Ratings. A J. D. Power and Associates vehicle quality survey asked new owners a

  1. a. Use a .05 level of significance and a test of independence to determine if a new owner’s vehicle quality rating is independent of the owner’s education. What is the p-value and what is your conclusion?
  2. b. Use the overall percentage of average, outstanding, and exceptional ratings to comment upon how new owners rate the quality of their recently purchased automobiles.

a.

Expert Solution
Check Mark
To determine

Perform a hypothesis test to check whether a new owner’s vehicle quality rating is independent of the owner’s education at 0.05 level of significance.

Find the p-value and draw the conclusion of the study.

Answer to Problem 12E

The p-value is 0.3624.

The new owner’s vehicle quality rating is not independent of the owner’s education.

Explanation of Solution

Calculation:

The given information shows that the sample of 500 automobile owners provides information regarding the level of education with level of quality rating of owners.

State the test hypotheses:

Null hypothesis:

 H0:The two cataegorical variables are independent.

That is, quality rating is independent of the education of the owner.

Alternative hypothesis:

 Ha:The two cataegorical variables are not independent.

That is, quality rating is not independent of the education of the owner.

The row and column totals are tabulated below:

Quality ratingSome HSHS GradSome CollegeCollege GradTotal
Average35302060145
Outstanding45455090230
Exceptional20253050125
Total100100100200500

The formula for expected frequency is given below:

eij=(Row i Total)(Coloumn j Total)Total Sample Size.

The expected frequency for each category is calculated as follows:

Quality ratingSome HSHS GradSome CollegeCollege Grade
Average(145)(100)500=29(145)(100)500=29(145)(100)500=29(145)(200)500=58
Outstanding(230)(100)500=46(230)(100)500=46(230)(100)500=46(230)(200)500=92
Exceptional(125)(100)500=25(125)(100)500=25(125)(100)500=25(125)(200)500=50

The formula for chi-square test statistic is given as,

χ2=i=1rj=1c(fijeij)2eij where (i,j)th element denotes the frequency corresponding to (i,j)th cell and r represent the rth row and c represents the cth column.

The value of chi-square test statistic is,

χ2={(3529)229+(3029)229+(2029)229+(6058)258+(4546)246+(4546)246+(5046)246+(9092)292+(2025)225+(2525)225+(3025)225+(5050)250}=1.24+0.03+2.79+0.07+0.02+0.02+0.35+0.04+1+0+1+0=6.57

Thus, the chi-square test statistic is 6.57.

Degrees of freedom:

The degrees of freedom is df=(r1)(c1) for the contingency table of r rows and c columns.

In the given problem r=3 and c=4.

Therefore,

df=(31)(41)=6

Level of significance:

The given level of significance is α=0.05.

p-value:

Software procedure:

Step -by-step software procedure to obtain p-value using EXCEL software:

  • Open an EXCEL sheet and select cell A1.
  • In cell A1 enter the formula =CHISQ.DIST.RT(6.57,6).
  • Press Enter.
  • Output using EXCEL software is given below:

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List), Chapter 12.2, Problem 12E

From the EXCEL output, the p-value is 0.3624.

Rejection rule:

  • If the p-valueα, then reject the null hypothesis.
  • Otherwise, failed to reject the null hypothesis.

Conclusion:

Here, the p-value is greater than the level of significance.

That is, p-value(=0.3624)>α(=0.05)

Thus, the decision is “fail to reject the null hypothesis”.

Therefore, the data do not provide sufficient evidence to conclude that the column variables are not independent of row variables.

That is, there is no association between column variable and row variable. In addition, quality ratings does not differ with the education of the owner.

Thus, the data do not provide sufficient evidence to conclude that the new owner’s vehicle quality rating is not independent of the owner’s education.

b.

Expert Solution
Check Mark
To determine

Find the overall percentage of average, outstanding and exceptional ratings and comment how new owners rate the quality of their recently purchased automobiles.

Explanation of Solution

Calculation:

The overall percentage of average rating is,

145500×100=29%.

The overall percentage of outstanding rating is,

230500×100=46%.

The overall percentage of exceptional rating is,

125500×100=25%.

Thus, the new owners are satisfied with almost 50% outstanding quality rating and 46%+25%=69% outstanding or exceptional rating for their new automobiles.

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Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

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