In fitting a least squares line ton= 15 data points, the following quantities were computed: SSxx = 55, SSyy = 198, SSxy = − 88,
a. Find the least squares line.
b. Graph the least squares line.
c. Calculate SSE.
d. Calculate s2.
e. Find a 90% confidence interval for β1. Interpret this estimate.
f. Find a 90% confidence interval for the mean value of y when x = 15.
g. Find a 90% prediction interval for y when x = 15.
a.
To find: The least square line.
Answer to Problem 11.101LM
The fitted least square line is
Explanation of Solution
Given info: The values are
Calculation:
The least square line is obtained below,
Where,
Substitute the value
The value of
Therefore, the value of
Substitute the values
The value of
The value of
The fitted regression line is,
Therefore, the fitted least square line is
b.
To plot: The line
Answer to Problem 11.101LM
The plotted line is,
Explanation of Solution
Calculations:
To plot the line on the graph, first to obtain the at least two points on the line
From the information, the line is
The two points are obtained below,
Let
Substitute the x value in equation (1) to get the value of y.
Therefore,
Let
Substitute the x value in equation (1) to get the value of y.
Therefore,
Based on the points
- 1. From the graph locate the value 1 on X-axis and 35.48 on Y-axis and mark the intersection point.
- 2. Locate the value 5 on X-axis and 29.08 on Y-axis and mark the intersection point.
- 3. Draw the line by combining the both intersection points.
- 4. The Graph is shown below.
c.
To find: The value of SSE.
Answer to Problem 11.101LM
Explanation of Solution
Calculations:
The formula for SSE is obtained below:
Substitute
The value of SSE is,
Therefore, the value of SSE is 57.2.
d.
To find: The value of
Answer to Problem 11.101LM
Explanation of Solution
Calculation:
Formula for
Substitute 57.2 for SSE and 15 for n in the formula.
The value of
Therefore, the value of
e.
To find: The 90% confidence interval for the slope
To interpret: The results.
Answer to Problem 11.101LM
The 90% confidence interval for the slope
Explanation of Solution
Calculation:
The formula for the confidence interval for estimate of
Where,
For confidence coefficient is 0.90. So that level of significance
The degrees of freedom is,
From table III Appendix D:
- 1. In the column locate the df as 13.
- 2. In the row locate the level of significance at 0.05.
- 3. The row and column of intersection point is 1.771 is the critical value.
Substitute
The 90% confidence interval for the slope
The 90% confidence interval for the slope
Interpretation:
It can be expected that the 90% confidence that each additional unit change in the mean value of y for each unit change in the x value is between –2.10 and –1.10.
f.
To find: The 90% confidence interval for the mean value of y when
Answer to Problem 11.101LM
The 90% confidence interval is.
Explanation of Solution
Calculations:
The formula for the 90% confidence interval is obtained below:
The estimated value for
The estimated value for
Substitute
The 90% confidence interval is,
Therefore, the 90% confidence interval is
g.
To find: The 90% prediction interval for the mean value of y when
Answer to Problem 11.101LM
Explanation of Solution
Calculations:
Formula for the 90% confidence interval is obtained below:
The estimated value for
The estimated value for
Substitute
The 90% prediction interval is,
Therefore, the 90% prediction interval is
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Chapter 11 Solutions
EBK STATISTICS FOR BUSINESS AND ECONOMI
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