Concept explainers
Find the values of k, which correspond to the useful model at the 0.05 level of significance.
Explain the large value of
Answer to Problem 32E
The values of k less than 9 corresponds to the useful model at the 0.05 level of significance.
Explanation of Solution
Calculation:
It is given that the sample size (n) is 15 and the value of
Test statistic:
Where, n is the sample size and k is the number of variables in the model.
The value of F test statistic is calculated as follows:
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph >
Probability Distribution Plot choose View Probability > OK. - From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 1 and Denominator df as 13.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 4.667.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 117 is greater than the critical value of 4.667.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 2 and Denominator df as 12.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 3.885.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 54 is greater than the critical value of 3.885.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 3 and Denominator df as 11.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 3.587.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 33 is greater than the critical value of 3.587.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 3 and Denominator df as 11.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 3.587.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 33 is greater than the critical value of 3.587.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 5 and Denominator df as 9.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 3.478.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 16.2 is greater than the critical value of 3.478.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 6 and Denominator df as 8.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 3.482.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 12 is greater than the critical value of 3.482.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 7 and Denominator df as 7.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 3.581.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 9 is greater than the critical value of 3.581.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 8 and Denominator df as 6.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 3.787.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 6.75 is greater than the critical value of 3.787.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 9 and Denominator df as 5.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 4.772.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 5 is greater than the critical value of 4.772.
Hence, reject the null hypothesis.
Thus, there is convincing evidence that the model is useful.
For
The value of F test statistic is calculated as follows:
P-value:
Software procedure:
Step-by-step procedure to find the P-value using the MINITAB software:
- Choose Graph > Probability Distribution Plot choose View Probability > OK.
- From Distribution, choose ‘F’ distribution.
- Enter the Numerator df as 10 and Denominator df as 4.
- Click the Shaded Area tab.
- Choose Probability Value and Right Tail for the region of the curve to shade.
- Enter the Probability value as 0.05.
- Click OK.
Output obtained using the MINITAB software is represented as follows:
From the MINITAB output, the critical value is 5.964.
Conclusion:
If the F test statistic value is greater than the critical value, then reject the null hypothesis.
Therefore, the F test statistic of 3.6 is less than the critical value of 5.964.
Hence, fail to reject the null hypothesis.
Thus, there is convincing evidence that the model is not useful.
Conclusion:
For the value of k less than 9, there is evidence that the model is useful. For
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Chapter 14 Solutions
Introduction to Statistics and Data Analysis
- Find the equation of the regression line for the following data set. x 1 2 3 y 0 3 4arrow_forwardOlympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?arrow_forward
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