
(a)
To calculate:
The sum of squared errors, SSE.

Answer to Problem 9CR
Solution:
The required SSE is 16.7246.
Explanation of Solution
Given Information:
The following data is collected on the number of years of post high-school education and the annual incomes of eight people ten years after graduation from high school.
Thread Count | 150 | 200 | 225 | 250 | 275 | 300 | 350 | 400 |
Price (in Dollars) | 18 | 21 | 25 | 28 | 30 | 31 | 35 | 45 |
The least Squares regression line is the line for which the average variation from the data is the smallest, also called the line of best fit, given by
Where
And
Formula used:
The equation of least-squares regression line is given by,
Where
And
Where n is the number of data pairs in the sample,
And
The sum of squared errors (SSE) for a regression line is calculated as,
Where,
And
Calculation:
Thread Count | Price(in Dollars) | |||
150 | 18 | 2700 | 22500 | 324 |
200 | 21 | 4200 | 40000 | 441 |
225 | 25 | 5625 | 50625 | 625 |
250 | 28 | 7000 | 62500 | 784 |
275 | 30 | 8250 | 75625 | 900 |
300 | 31 | 9300 | 90000 | 961 |
350 | 35 | 12250 | 122500 | 1225 |
400 | 45 | 18000 | 160000 | 2025 |
Let
And
The slope of the least-squares regression line is calculated as,
Where,
Substitute 150 for
Proceed in the same manner to calculate
The slope of the least-squares regression line is calculated as,
Substitute 2150 for
The y-intercept of regression line is calculated as,
Substitute 2150 for
The equation of least-squares regression line is given by,
Substitute 28.6514 for
Number of years |
Annual income |
Predicted value |
||
150 | 18 | 16.965 | 1.035 | 1.071225 |
200 | 21 | 22.085 | -1.085 | 1.177225 |
225 | 25 | 24.645 | 0.355 | 0.126025 |
250 | 28 | 27.205 | 0.795 | 0.632025 |
275 | 30 | 29.765 | 0.235 | 0.055225 |
300 | 31 | 32.325 | -1.325 | 1.755625 |
350 | 35 | 37.445 | -2.445 | 5.978025 |
400 | 45 | 42.565 | 2.435 | 5.929225 |
The predicted values are calculated as,
The predicted value
Substitute 150 for
Proceed in the same manner to calculate
The residual is calculated as,
Substitute 18 for
Square both sides of the equation.
Proceed in the same manner to calculate
Conclusion:
Thus, the SSE is 16.7246
(b)
To calculate:
The standard error of estimate,

Answer to Problem 9CR
Solution:
The required standard error of estimate is
Explanation of Solution
Given Information:
The following data is collected on the number of years of post high-school education and the annual incomes of eight people ten years after graduation from high school.
Thread Count | 150 | 200 | 225 | 250 | 275 | 300 | 350 | 400 |
Price (in Dollars) | 18 | 21 | 25 | 28 | 30 | 31 | 35 | 45 |
Formula used:
The standard error of estimate, which is used to measure by how much the sample data points deviate from regression line is given by,
Where,
n is the number of data pairs in the sample,
And SSE is the sum of squared errors.
Calculation:
The standard error of estimate is calculated as,
Substitute 5868.153 for SSE and 8 for n in the above formula.
Conclusion:
Thus, the standard error of estimate is
(c)
The 95% prediction interval for the price of 350-thread count sheets.

Answer to Problem 9CR
Solution:
The required prediction interval is.
Explanation of Solution
Given Information:
The following data is collected on the number of years of post high-school education and the annual incomes of eight people ten years after graduation from high school.
Thread Count | 150 | 200 | 225 | 250 | 275 | 300 | 350 | 400 |
Price (in Dollars) | 18 | 21 | 25 | 28 | 30 | 31 | 35 | 45 |
Formula used:
The margin of error of a prediction interval for an individual y-value is calculated as,
With degree of freedom
Where,
n is the number of data pairs in the sample,
SSE is the sum of squared errors,
And
Then the prediction interval for an individual y-value is,
Calculation:
It is given that the level of prediction is 0.95 then the level of significance is calculated as,
Then,
The mean of the number of years of post high school education is calculated as,
Substitute 2150 for
The margin of error of a prediction interval for an individual y-value is calculated as,
Substitute 2.447 for
The
The regression line is,
Substitute 350 for
The prediction interval is,
Conclusion:
The required prediction interval is.
(d)
The 95% confidence interval for the y-intercept of the regression line.

Answer to Problem 9CR
Solution:
The required confidence interval is
Explanation of Solution
Given Information:
The following data is collected on the number of years of post high-school education and the annual incomes of eight people ten years after graduation from high school.
Thread Count | 150 | 200 | 225 | 250 | 275 | 300 | 350 | 400 |
Price (in Dollars) | 18 | 21 | 25 | 28 | 30 | 31 | 35 | 45 |
Formula Used:
The
Coefficient of determination measures the proportion of variation in the response variable caused by explanatory variable which is simply the square of r, the correlation coefficient.
The standard error of estimate,
In ANOVA,
Grand Mean is the weighted mean of the
Sum of Squares among Treatments (SST) is the measures the variation between the sample means and the grand mean, given by,
Sum of Squares for Error (SSE) is the measures the variation in the sample data resulting from the variability within each sample,
Total Variation, it is the sum of the squared deviations from the grand mean for all of the data values in each sample, given by
Mean Square for Treatments (MST) found by dividing the sum of squares among treatments by its degrees of freedom, given by
Mean Square for Error (MSE) found by dividing the sum of squares for error by its degrees of freedom, given by
Test Statistic for an ANOVA Test is used when independent, simple random samples are taken from populations with variances that are unknown and assumed to be equal, where all of the
Calculation:
To generate the regression table in excel follow the given steps:
1. Under data tab, choose data analytics and then select regression.
2. Select the input Y range and enter the range of the given
3.Choose 95% confidence interval and click OK.
The following table will appear.
Regression Statistics | |
Multiple R | 0.983094904 |
R Square | 0.96647559 |
Adjusted R Square | 0.960888189 |
Standard Error | 1.66955532 |
Observations | 8 |
ANOVA | |||||
df | SS | MS | F | Significance F | |
Regression | 1 | 482.1505102 | 482.1505102 | 172.9740696 | 1.19253E-05 |
Residual | 6 | 16.7244898 | 2.787414966 | ||
Total | 7 | 498.875 |
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 1.591836735 | 2.17509095 | 0.73184835 | 0.491845191 | -3.730419077 | 6.914092547 |
Slope | 0.10244898 | 0.007789635 | 13.15196067 | 1.19253E-05 | 0.083388428 | 0.121509531 |
RESIDUAL OUTPUT | |||
Observation | Predicted y | Residuals | Standard Residuals |
1 | 16.95918367 | 1.040816327 | 0.673359012 |
2 | 22.08163265 | -1.081632653 | -0.699765248 |
3 | 24.64285714 | 0.357142857 | 0.231054563 |
4 | 27.20408163 | 0.795918367 | 0.514921598 |
5 | 29.76530612 | 0.234693878 | 0.151835856 |
6 | 32.32653061 | -1.326530612 | -0.858202663 |
7 | 37.44897959 | -2.448979592 | -1.584374147 |
8 | 42.57142857 | 2.428571429 | 1.571171029 |
The confidence interval of the y-intercept can be constructed by adding and subtracting the margin of error to the point estimate by using Microsoft excel.
Referring regression statistics,
Standard error is the standard error of estimate,
The lower 95% and the upper 95% gives the confidence interval of the y-intercept.
The intercept given in the row of the table above is the
So the regression line is,
The lower and the upper endpoints for a 95% confidence interval for the y-intercept of the regression line,
Conclusion:
Thus, the 95% confidence interval for the y-intercept of the regression line is.
(e)
Construct a 95% confidence interval for the slope of the regression line.

Answer to Problem 9CR
Solution:
The required confidence interval is
Explanation of Solution
Given Information:
The following data is collected on the number of years of post high-school education and the annual incomes of eight people ten years after graduation from high school.
Thread Count | 150 | 200 | 225 | 250 | 275 | 300 | 350 | 400 |
Price (in Dollars) | 18 | 21 | 25 | 28 | 30 | 31 | 35 | 45 |
Formula Used:
The
Coefficient of determination measures the proportion of variation in the response variable caused by explanatory variable which is simply the square of r, the correlation coefficient.
The standard error of estimate,
In ANOVA,
Grand Mean is the weighted mean of the
Sum of Squares among Treatments (SST) is the measures the variation between the sample means and the grand mean, given by,
Sum of Squares for Error (SSE) is the measures the variation in the sample data resulting from the variability within each sample,
Total Variation, it is the sum of the squared deviations from the grand mean for all of the data values in each sample, given by
Mean Square for Treatments (MST) found by dividing the sum of squares among treatments by its degrees of freedom, given by
Mean Square for Error (MSE) found by dividing the sum of squares for error by its degrees of freedom, given by
Test Statistic for an ANOVA Test is used when independent, simple random samples are taken from populations with variances that are unknown and assumed to be equal, where all of the
Calculation:
To generate the regression table in excel follow the given steps:
1. Under data tab, choose data analytics and then select regression.
2. Select the input Y range and enter the range of the given
3.Choose 95% confidence interval and click OK.
The following table will appear.
Regression Statistics | |||||||||||||||||
Multiple R | 0.983094904 | ||||||||||||||||
R Square | 0.96647559 | ||||||||||||||||
Adjusted R Square | 0.960888189 | ||||||||||||||||
Standard Error | 1.66955532 | ||||||||||||||||
Observations | 8 | ||||||||||||||||
ANOVA | |||||||||||||||||
df | SS | MS | F | Significance F | |||||||||||||
Regression | 1 | 482.1505102 | 482.1505102 | 172.9740696 | 1.19253E-05 | ||||||||||||
Residual | 6 | 16.7244898 | 2.787414966 | ||||||||||||||
Total | 7 | 498.875 | |||||||||||||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | ||||||||||||
Intercept | 1.591836735 | 2.17509095 | 0.731848356 | 0.491845191 | -3.730419077 | 6.914092547 | |||||||||||
Slope | 0.10244898 | 0.007789635 | 13.15196067 | 1.19253E-05 | 0.083388428 | 0.121509531 | |||||||||||
RESIDUAL OUTPUT | |||||||||||||||||
Observation | Predicted y | Residuals | Standard Residuals | ||||||||||||||
1 | 16.95918367 | 1.040816327 | 0.673359012 | ||||||||||||||
2 | 22.08163265 | -1.081632653 | -0.699765248 | ||||||||||||||
3 | 24.64285714 | 0.357142857 | 0.231054563 | ||||||||||||||
4 | 27.20408163 | 0.795918367 | 0.514921598 | ||||||||||||||
5 | 29.76530612 | 0.234693878 | 0.151835856 | ||||||||||||||
6 | 32.32653061 | -1.326530612 | -0.858202663 | ||||||||||||||
7 | 37.44897959 | -2.448979592 | -1.584374147 | ||||||||||||||
8 | 42.57142857 | 2.428571429 | 1.571171029 | ||||||||||||||
The lower 95% and the upper 95% gives the confidence interval of the slope.
The intercept given in the row of the table above is the
So the regression line is,
The lower and the upper endpoints for a 95% confidence interval for the slope of the regression line,
Conclusion:
Thus, the 95% confidence interval for the slope of the regression line is
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Chapter 12 Solutions
Beginning Statistics, 2nd Edition
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