Concept explainers
Problem 9 examined the relationship between weight and income for a sample of n = 10 women. Weights were classified in five categories and had a
- a. Find the regression equation for predicting income from weight. (Identify the income scores as X values and the weight scores as Y values.)
- b. What percentage of the variance in the income is accounted for by the regression equation? (Compute the
correlation , r. then find r2.) - c. Does the regression equation account for a significant portion of the variance in income? Use α= .05 to evaluate the F-ratio.
Weight (X) | Income (Y) |
1 | 125 |
2 | 78 |
4 | 49 |
3 | 63 |
5 | 35 |
2 | 84 |
5 | 38 |
3 | 51 |
1 | 93 |
4 | 44 |
a.
Answer to Problem 25P
The regression equation of the data is
Explanation of Solution
Given info: For 10 women weights are classified in five categories with mean of
Weight(X) | Income(Y) |
1 | 125 |
2 | 78 |
4 | 49 |
3 | 63 |
5 | 35 |
2 | 84 |
5 | 38 |
3 | 51 |
1 | 93 |
4 | 44 |
Calculation:
Generally the linear regression equation is defined as
Formula to calculate
Formula to calculate the
Mean of the variable
Thus, the mean of the variable
Mean of the variable
Thus the mean of the variable y is 66.
For calculating
Sr.no |
|
|
|
|
|
|
|
1 | 1 | 125 | 12 | 59 | 4 | 3481 | -118 |
2 | 2 | 78 | -1 | 12 | 1 | 144 | -12 |
3 | 4 | 49 | 1 | -17 | 1 | 289 | -17 |
4 | 3 | 63 | 0 | -3 | 0 | 9 | 0 |
5 | 5 | 35 | 2 | -31 | 4 | 961 | -62 |
6 | 2 | 84 | -1 | 18 | 1 | 324 | -18 |
7 | 5 | 38 | 2 | -28 | 4 | 784 | -56 |
8 | 3 | 51 | 0 | -15 | 0 | 225 | 0 |
9 | 1 | 93 | -2 | 27 | 4 | 729 | -54 |
10 | 4 | 44 | 1 | -22 | 1 | 484 | -22 |
Total | 0 | 0 | 20 | 7430 | -359 |
Where,
Substitute
Substitute
Thus, the regression equation of the data is
b.
Answer to Problem 25P
The percentage of variance in the income accounted for the regression equation is 86.7%.
Explanation of Solution
Calculation:
Formula to calculate the Pearson correlation is,
Where covariability of the variable x and y is
As calculated in part a
Substitute
Thus, the Pearson correlation of the given data is
Thus, 86.7% of the variance in the income is accounted for by the regression equation.
c.
Answer to Problem 25P
The regression equation is significance at 5% level of significance.
Explanation of Solution
Calculation:
As calculated in part a
Set the null hypothesis
The alternative hypothesis
Formula to calculate the
Where,
Substitute
Substitute
Thus the
Thus the
Thus the is 123.523.
Substitute 6441.81 for
Thus the F-ratio is 76.437.
From the F-table, the value of
Decision rule:
If the calculated F-value is less than table F-value, then there is enough evidence to accept to reject the null hypothesis.
As
Thus, the regression equation is significance at 5% level of significance.
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Chapter 14 Solutions
EBK ESSENTIALS OF STATISTICS FOR THE BE
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