The article first introduced in Exercise 13.34 of Section 13.3 gave data on the dimensions of 27 representative food products.
- a. Use the data set given there to test the hypothesis that there is a positive linear relationship between x = minimum width and y = maximum width of an object.
- b. Calculate and interpret se.
- c. Calculate a 95% confidence interval for the
mean maximum width of products with a minimum width of 6 cm. - d. Calculate a 95% prediction interval for the maximum width of a food package with a minimum width of 6 cm.
13.34 The article “Vital Dimensions in Volume Perception: Can the Eye Fool the Stomach?” (Journal of Marketing Research [1999]: 313–326) gave the accompanying data on the dimensions of 27 representative food products (Gerber baby food, Cheez Whiz, Skippy Peanut Butter, and Ahmed’s tandoori paste, to name a few).
- a. Fit the simple linear regression model that would allow prediction of the maximum width of a food container based on its minimum width.
- b. Calculate the standardized residuals (or just the residuals if a computer program that doesn’t give standardized residuals is used) and make a residual plot to determine whether there are any outliers.
- c. The data point with the largest residual is for a 1-liter Coke bottle. Delete this data point and determine the equation of the regression line. Did deletion of this point result in a large change in the equation of the estimated regression line?
- d. For the regression line of Part (c), interpret the estimated slope and, if appropriate, the intercept.
- e. For the data set with the Coke bottle deleted, are the assumptions of the simple linear regression model reasonable? Give statistical evidence.
a.
Check whether there is a positive linear relationship between minimum and maximum width of an object.
Answer to Problem 44E
There is convincing evidence that there is a positive linear relationship between minimum and maximum width of an object.
Explanation of Solution
Calculation:
The given data provide the dimensions of 27 representative food products.
1.
Here,
2.
Null hypothesis:
That is, there is no linear relationship between minimum and maximum width of an object.
3.
Alternative hypothesis:
That is, there is a positive linear relationship between minimum and maximum width of an object.
4.
Here, the significance level is
5.
Test Statistic:
The formula for test statistic is,
In the formula, b denotes the estimated slope,
6.
A standardized residual plot is shown below.
Standardized residual values and standardized residual plot:
Software procedure:
Step-by-step procedure to compute standardized residuals and its plot using MINITAB software:
- Select Stat > Regression > Regression > Fit Regression Model
- In Response, enter the column of Maximum width.
- In Continuous Predictors, enter the columns of Minimum width.
- In Graphs, select Standardized under Residuals for Plots.
- In Results, select for all observations under Fits and diagnostics.
- In Residuals versus the variables, select Minimum width.
- Click OK.
Output obtained the MINTAB software is given below:
From the standardized residual plot, it is observed that one point lies outside the horizontal band of 3 units from the central line of 0. The standardized residual for this outlier is 3.72, which is for the product 25.
Assumption:
Here, the assumption made is that, the simple linear regression model is appropriate for the data, even though there is one extreme standardized residual.
7.
Calculation:
Test Statistic:
In the MINITAB output, the test statistic value is displayed in the column “T-value” corresponding to “Minimum width”, in the section “Coefficients”. The value is 13.53.
8.
P-value:
From the above output, the correponding P-value is 0.
9.
Rejection rule:
If
Conclusion:
The P-value is 0.
The level of significance is 0.05.
The P-value is less than the level of significance.
That is,
Based on the rejection rule, reject the null hypothesis.
Thus, there is convincing evidence that there is a positive linear relationship between minimum and maximum width of an object.
b.
Compute and interpret
Answer to Problem 44E
Explanation of Solution
Calculation:
From the MINITAB output in Part (a), it is clear that
On an average, there is a 67.246% deviation of the maximum width in the sample from the value predicted by least-squares regression.
c.
Find the 95% confidence interval for the mean maximum width of products, for a minimum width of 6 cm.
Answer to Problem 44E
The 95% interval for the mean maximum width of products, for a minimum width of 6 cm is (5.708, 6.647).
Explanation of Solution
Calculation:
The confidence interval for
From the MINITAB output in Part (a), the estimated linear regression line is
Point estimate:
The point estimate is calculated as follows.
Estimated standard deviation:
For the given x values, the summation values are given in the following table.
Minimum width (X) | |
1.8 | 3.24 |
2.7 | 7.29 |
2 | 4 |
2.6 | 6.76 |
3.15 | 9.9225 |
1.8 | 3.24 |
1.5 | 2.25 |
3.8 | 14.44 |
5 | 25 |
4.75 | 22.5625 |
2.8 | 7.84 |
2.1 | 4.41 |
2.2 | 4.84 |
2.6 | 6.76 |
2.6 | 6.76 |
2.9 | 8.41 |
5.1 | 26.01 |
10.2 | 104.04 |
3.5 | 12.25 |
1.2 | 1.44 |
1.7 | 2.89 |
1.75 | 3.0625 |
1.7 | 2.89 |
1.2 | 1.44 |
1.2 | 1.44 |
7.5 | 56.25 |
4.25 | 18.0625 |
The value of
Substitute,
Formula for Degrees of freedom:
The formula for degrees of freedom is,
The number of data values given is 27, that is
Critical value:
From the Appendix: Table of t Critical Values:
- Locate the value 25 in the degrees of freedom (df) column.
- Locate the 0.95 in the row of central area captured.
- The intersecting value that corresponds to the df 25 with confidence level 0.95 is 2.060.
Thus, the critical value for
Substitute,
Therefore, one can be 95% confident that the mean maximum width of products with a minimum width of 6 cm will be between 5.708 cm and 6.647 cm.
d.
Find a 95% prediction interval for the mean maximum width of products with a minimum width of 6 cm.
Answer to Problem 44E
The 95% prediction interval for the mean maximum width of products with a minimum width of 6 cm is (4.716, 7.640).
Explanation of Solution
Calculation:
The confidence interval for
The estimated standard deviation of the amount by which a single y observation deviates from the value predicted by an estimated regression line is,
Substitute
From Part (c), the critical value for
Substitute,
Therefore, the 95% prediction interval for the mean maximum width of products with a minimum width of 6 cm is (4.716, 7.640).
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Chapter 13 Solutions
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