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Concept explainers
a.
Obtain the null and the alternative hypotheses.
a.
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Explanation of Solution
Denote
The null and alternative hypotheses are given below:
Null Hypothesis
All the means of tuition rates for the various regions are equal.
Alternative Hypothesis
b.
Give the decision rule.
b.
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Explanation of Solution
The degrees of freedom for the numerator is obtained as given below:
The degrees of freedom for the denominator is obtained as given below:
The critical F value is as follows:
Here, the test is one-tailed and the level of significance
Use Table - B.6A Critical Values of the F Distribution (α = .05) to obtained critical F value.
- Locate 2 in the column of Degrees of Freedom for the Numerator
- Locate 11 in the row of Degrees of Freedom for the Denominator
- The intersection of row and column is 3.98
Thus, the critical F value is 3.98.
Decision rule:
If
Therefore, the decision rule is reject
c.
Obtain an ANOVA table.
Find the value of the test statistic.
c.
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Answer to Problem 3SR
The ANOVA table is,
Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F |
Treatments | 44.16 | 2 | 22.082 | 25.43 |
Error | 9.55 | 11 | 0.868 | |
Total | 53.71 | 13 |
The value of the test statistics is 25.43.
Explanation of Solution
Here, the level of significance
Step-by-step procedure to obtain ANOVA table using Excel-MegaStat:
- In EXCEL, Select Add-Ins > Mega Stat > Analysis of Variance.
- Choose One-Factor ANOVA.
- In Input
Range , enter the column of Northeast, Southeast and West. - Choose When p < 0.05.
- Click OK.
Output using the Excel-MegaStat software is given below:
From the output, the ANOVA table is obtained as given below:
Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F |
Treatments | 44.16 | 2 | 22.082 | 25.43 |
Error | 9.55 | 11 | 0.868 | |
Total | 53.71 | 13 |
From the output, the value of the test statistics is 25.43.
d.
Find the decision regarding the null hypothesis.
d.
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Answer to Problem 3SR
The null hypothesis is rejected.
Explanation of Solution
Decision:
The F value is 25.43 and the F critical value is 3.98.
Here, F value is greater than F critical value. That is, 25.43 > 3.98.
Using decision rule, reject the null hypothesis.
The decision is, reject the null hypothesis because the computed F value (25.43) is greater than the F critical value (3.81).
Thus, the null hypothesis
Therefore, there is sufficient evidence that there is a difference in the mean tuition rates for the various regions.
e.
Check if there be a significant difference between the mean tuition in the Northeast and that of the West.
Construct a 95% confidence interval for the difference.
e.
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Answer to Problem 3SR
There be a significant difference between the mean tuition in the Northeast and that of the West.
A 95% confidence interval for the difference is 2.9 and 5.5.
Explanation of Solution
A 95% confidence interval is as follows:
Where,
Step-by-step procedure to obtain ANOVA table using Excel-MegaStat:
- In EXCEL, Select Add-Ins > Mega Stat > Analysis of Variance.
- Choose One-Factor ANOVA.
- In Input Range, enter the column of Northeast, Southeast and West.
- In Post-Hoc Analysis, Choose When p < 0.05.
- Click OK.
Output using the Excel-MegaStat software is given below:
From the output, mean of northwest is 41, mean of west is 36.8, and MSE is 0.868.
The value of t is as follows:
Step-by-step procedure to obtain ANOVA table using Excel-MegaStat:
- In EXCEL, Select Add-Ins > MegaStat > Probability.
- Choose Continuous probability distributions.
- Select t-distribution and select calculate t given probability and enter probability as 0.05.
- Enter df as 11.
- Click Ok.
Output using the Excel-MegaStat software is given below:
From the output, the t is
Therefore, a 95% confidence interval for the difference is between 2.9 and 5.5. Both the end points are positive and does not include zero. Thus, there is a significant difference between the mean tuition rates in the Northeast and that of the West.
Thus, the 95% confidence interval for the difference is 2.9 and 5.5.
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Chapter 12 Solutions
Gen Combo Ll Statistical Techniques In Business And Economics; Connect Ac
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