13.5 Testing the Regression Coefficients
LEARNING OBJECTIVES
- Conduct and interpret a hypothesis test on individual regression coefficients.
Previously, we learned that the population model for the multiple regression equation is
[latex]\begin{eqnarray*}y&=&\beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_kx_k+\epsilon\end{eqnarray*}[/latex]
where [latex]x_1,x_2,\ldots,x_k[/latex] are the independent variables, [latex]\beta_0,\beta_1,\ldots,\beta_k[/latex] are the population parameters of the regression coefficients, and [latex]\epsilon[/latex] is the error variable. In multiple regression, we estimate each population regression coefficient [latex]\beta_i[/latex] with the sample regression coefficient [latex]b_i[/latex].
In the previous section, we learned how to conduct an overall model test to determine if the regression model is valid. If the outcome of the overall model test is that the model is valid, then at least one of the independent variables is related to the dependent variable—in other words, at least one of the regression coefficients [latex]\beta_i[/latex] is not zero. However, the overall model test does not tell us which independent variables are related to the dependent variable. To determine which independent variables are related to the dependent variable, we must test each of the regression coefficients.
Testing the Regression Coefficients
For an individual regression coefficient, we want to test if there is a relationship between the dependent variable [latex]y[/latex] and the independent variable [latex]x_i[/latex].
- No Relationship. There is no relationship between the dependent variable [latex]y[/latex] and the independent variable [latex]x_i[/latex]. In this case, the regression coefficient [latex]\beta_i[/latex] is zero. This is the claim for the null hypothesis in an individual regression coefficient test: [latex]H_0:\beta_i=0[/latex].
- Relationship. There is a relationship between the dependent variable [latex]y[/latex] and the independent variable [latex]x_i[/latex]. In this case, the regression coefficients [latex]\beta_i[/latex] is not zero. This is the claim for the alternative hypothesis in an individual regression coefficient test: [latex]H_a:\beta_i\neq 0[/latex]. We are not interested if the regression coefficient [latex]\beta_i[/latex] is positive or negative, only that it is not zero. We only need to find out if the regression coefficient is not zero to demonstrate that there is a relationship between the dependent variable and the independent variable. This makes the test on a regression coefficient a two-tailed test.
In order to conduct a hypothesis test on an individual regression coefficient [latex]\beta_i[/latex], we need to use the distribution of the sample regression coefficient [latex]b_i[/latex]:
- The mean of the distribution of the sample regression coefficient is the population regression coefficient [latex]\beta_i[/latex].
- The standard deviation of the distribution of the sample regression coefficient is [latex]\sigma_{b_i}[/latex]. Because we do not know the population standard deviation, we must estimate [latex]\sigma_{b_i}[/latex] with the sample standard deviation [latex]s_{b_i}[/latex].
- The distribution of the sample regression coefficient follows a normal distribution.
Conducting a Hypothesis Test on a Regression Coefficient
Follow these steps to perform a hypothesis test on an individual regression coefficient:
- Write down the null hypothesis that there is no relationship between the dependent variable [latex]y[/latex] and the independent variable [latex]x_i[/latex]:
[latex]\begin{eqnarray*}H_0:&&\beta_i=0\\\\\end{eqnarray*}[/latex]
- Write down the alternative hypotheses that there is a relationship between the dependent variable [latex]y[/latex]and the independent variable [latex]x_i[/latex]:
[latex]\begin{eqnarray*}H_a:&&\beta_i\neq 0\\\\\end{eqnarray*}[/latex]
- Collect the sample information for the test and identify the significance level [latex]\alpha[/latex].
- The [latex]p-\text{value}[/latex] is the sum of the area in the tails of the [latex]t[/latex]-distribution. The [latex]t[/latex]-score and degrees of freedom are
[latex]\begin{eqnarray*}t&=&\frac{b_i-\beta_i}{s_{b_i}}\\\\df&=&n-k-1\\\\\end{eqnarray*}[/latex]
- Compare the [latex]p-\text{value}[/latex] to the significance level and state the outcome of the test.
- If [latex]p-\text{value}\leq\alpha[/latex], reject [latex]H_0[/latex] in favour of [latex]H_a[/latex].
- The results of the sample data are significant. There is sufficient evidence to conclude that the null hypothesis [latex]H_0[/latex] is an incorrect belief and that the alternative hypothesis [latex]H_a[/latex] is most likely correct.
- If [latex]p-\text{value}\gt\alpha[/latex], do not reject [latex]H_0[/latex].
- The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis [latex]H_a[/latex] may be correct.
- If [latex]p-\text{value}\leq\alpha[/latex], reject [latex]H_0[/latex] in favour of [latex]H_a[/latex].
- Write down a concluding sentence specific to the context of the question.
The required [latex]t[/latex]-score and [latex]p-\text{value}[/latex] for the test can be found on the regression summary table, which we learned how to generate in Excel in a previous section.
EXAMPLE
The human resources department at a large company wants to develop a model to predict an employee’s job satisfaction from the number of hours of unpaid work per week the employee does, the employee’s age, and the employee’s income. A sample of [latex]25[/latex] employees at the company is taken, and the data is recorded in the table below. The employee’s income is recorded in [latex]\$1000[/latex]s, and the job satisfaction score is out of [latex]10[/latex], with higher values indicating greater job satisfaction.
| Job Satisfaction | Hours of Unpaid Work per Week | Age | Income ([latex]\$1000[/latex]s) |
|---|---|---|---|
| 4 | 3 | 23 | 60 |
| 5 | 8 | 32 | 114 |
| 2 | 9 | 28 | 45 |
| 6 | 4 | 60 | 187 |
| 7 | 3 | 62 | 175 |
| 8 | 1 | 43 | 125 |
| 7 | 6 | 60 | 93 |
| 3 | 3 | 37 | 57 |
| 5 | 2 | 24 | 47 |
| 5 | 5 | 64 | 128 |
| 7 | 2 | 28 | 66 |
| 8 | 1 | 66 | 146 |
| 5 | 7 | 35 | 89 |
| 2 | 5 | 37 | 56 |
| 4 | 0 | 59 | 65 |
| 6 | 2 | 32 | 95 |
| 5 | 6 | 76 | 82 |
| 7 | 5 | 25 | 90 |
| 9 | 0 | 55 | 137 |
| 8 | 3 | 34 | 91 |
| 7 | 5 | 54 | 184 |
| 9 | 1 | 57 | 60 |
| 7 | 0 | 68 | 39 |
| 10 | 2 | 66 | 187 |
| 5 | 0 | 50 | 49 |
Previously, we found the multiple regression equation to predict the job satisfaction score from the other variables:
[latex]\begin{eqnarray*}\hat{y}&=&4.7993-0.3818x_1+0.0046x_2+0.0233x_3\\\\\hat{y}&=&\text{predicted job satisfaction score}\\x_1&=&\text{hours of unpaid work per week}\\x_2&=&\text{age}\\x_3&=&\text{income (\$1000s)}\end{eqnarray*}[/latex]
At the [latex]5\%[/latex] significance level, test the relationship between the dependent variable “job satisfaction” and the independent variable “hours of unpaid work per week”.
Solution
Hypotheses:
[latex]\begin{eqnarray*}H_0:&&\beta_1=0\\H_a:&&\beta_1\neq 0\end{eqnarray*}[/latex]
[latex]p-\text{value}[/latex]:
The regression summary table generated by Excel is shown below:
| SUMMARY OUTPUT | ||||||
|---|---|---|---|---|---|---|
| Regression Statistics | ||||||
| Multiple R | 0.711779225 | |||||
| R Square | 0.506629665 | |||||
| Adjusted R Square | 0.436148189 | |||||
| Standard Error | 1.585212784 | |||||
| Observations | 25 | |||||
| ANOVA | ||||||
| df | SS | MS | F | Significance F | ||
| Regression | 3 | 54.189109 | 18.06303633 | 7.18812504 | 0.001683189 | |
| Residual | 21 | 52.770891 | 2.512899571 | |||
| Total | 24 | 106.96 | ||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
| Intercept | 4.799258185 | 1.197185164 | 4.008785216 | 0.00063622 | 2.309575344 | 7.288941027 |
| Hours of Unpaid Work per Week | -0.38184722 | 0.130750479 | -2.9204269 | 0.008177146 | -0.65375772 | -0.10993671 |
| Age | 0.004555815 | 0.022855709 | 0.199329423 | 0.843922453 | -0.04297523 | 0.052086864 |
| Income ([latex]\$1000[/latex]s) | 0.023250418 | 0.007610353 | 3.055103771 | 0.006012895 | 0.007423823 | 0.039077013 |
The [latex]p-\text{value}[/latex] for the test on the hours of unpaid work per week regression coefficient is in the bottom part of the table under the P-value column of the Hours of Unpaid Work per Week row. So the [latex]p-\text{value}=0.0082[/latex].
Conclusion:
Because [latex]p-\text{value}=0.0082\lt 0.05=\alpha[/latex], we reject the null hypothesis in favour of the alternative hypothesis. At the [latex]5\%[/latex] significance level, there is enough evidence to suggest that there is a relationship between the dependent variable “job satisfaction” and the independent variable “hours of unpaid work per week.”
NOTES
- The null hypothesis [latex]\beta_1=0[/latex] is the claim that the regression coefficient for the independent variable [latex]x_1[/latex] is zero. That is, the null hypothesis is the claim that there is no relationship between the dependent variable and the independent variable, “hours of unpaid work per week.”
- The alternative hypothesis is the claim that the regression coefficient for the independent variable [latex]x_1[/latex] is not zero. The alternative hypothesis is the claim that there is a relationship between the dependent variable and the independent variable, “hours of unpaid work per week.”
- When conducting a test on a regression coefficient, make sure to use the correct subscript on [latex]\beta[/latex] to correspond to how the independent variables were defined in the regression model and which independent variable is being tested. Here the subscript on [latex]\beta[/latex] is [latex]1[/latex] because the “hours of unpaid work per week” is defined as [latex]x_1[/latex] in the regression model.
- The [latex]p-\text{value}[/latex] for the tests on the regression coefficients are located in the bottom part of the table under the P-value column heading in the corresponding independent variable row.
- Because the alternative hypothesis is a [latex]\neq[/latex], the [latex]p-\text{value}[/latex] is the sum of the area in the tails of the [latex]t[/latex]-distribution. This is the value calculated out by Excel in the regression summary table.
- The [latex]p-\text{value}[/latex] of [latex]0.0082[/latex] is a small probability compared to the significance level and so is unlikely to happen assuming the null hypothesis is true. This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis. In other words, the regression coefficient [latex]\beta_1[/latex] is not zero, and so there is a relationship between the dependent variable “job satisfaction” and the independent variable “hours of unpaid work per week.” This means that the independent variable “hours of unpaid work per week” is useful in predicting the dependent variable.
EXAMPLE
The human resources department at a large company wants to develop a model to predict an employee’s job satisfaction from the number of hours of unpaid work per week the employee does, the employee’s age, and the employee’s income. A sample of [latex]25[/latex] employees at the company is taken, and the data is recorded in the table below. The employee’s income is recorded in [latex]\$1000[/latex]s, and the job satisfaction score is out of [latex]10[/latex], with higher values indicating greater job satisfaction.
| Job Satisfaction | Hours of Unpaid Work per Week | Age | Income ([latex]\$1000[/latex]s) |
|---|---|---|---|
| 4 | 3 | 23 | 60 |
| 5 | 8 | 32 | 114 |
| 2 | 9 | 28 | 45 |
| 6 | 4 | 60 | 187 |
| 7 | 3 | 62 | 175 |
| 8 | 1 | 43 | 125 |
| 7 | 6 | 60 | 93 |
| 3 | 3 | 37 | 57 |
| 5 | 2 | 24 | 47 |
| 5 | 5 | 64 | 128 |
| 7 | 2 | 28 | 66 |
| 8 | 1 | 66 | 146 |
| 5 | 7 | 35 | 89 |
| 2 | 5 | 37 | 56 |
| 4 | 0 | 59 | 65 |
| 6 | 2 | 32 | 95 |
| 5 | 6 | 76 | 82 |
| 7 | 5 | 25 | 90 |
| 9 | 0 | 55 | 137 |
| 8 | 3 | 34 | 91 |
| 7 | 5 | 54 | 184 |
| 9 | 1 | 57 | 60 |
| 7 | 0 | 68 | 39 |
| 10 | 2 | 66 | 187 |
| 5 | 0 | 50 | 49 |
Previously, we found the multiple regression equation to predict the job satisfaction score from the other variables:
[latex]\begin{eqnarray*}\hat{y}&=&4.7993-0.3818x_1+0.0046x_2+0.0233x_3\\\\\hat{y}&=&\text{predicted job satisfaction score}\\x_1&=&\text{hours of unpaid work per week}\\x_2&=&\text{age}\\x_3&=&\text{income (\$1000s)}\end{eqnarray*}[/latex]
At the [latex]5\%[/latex] significance level, test the relationship between the dependent variable “job satisfaction” and the independent variable “age”.
Solution
Hypotheses:
[latex]\begin{eqnarray*}H_0:&&\beta_2=0\\H_a:&&\beta_2\neq 0\end{eqnarray*}[/latex]
[latex]p-\text{value}[/latex]:
The regression summary table generated by Excel is shown below:
| SUMMARY OUTPUT | ||||||
| Regression Statistics | ||||||
| Multiple R | 0.711779225 | |||||
| R Square | 0.506629665 | |||||
| Adjusted R Square | 0.436148189 | |||||
| Standard Error | 1.585212784 | |||||
| Observations | 25 | |||||
| ANOVA | ||||||
| df | SS | MS | F | Significance F | ||
| Regression | 3 | 54.189109 | 18.06303633 | 7.18812504 | 0.001683189 | |
| Residual | 21 | 52.770891 | 2.512899571 | |||
| Total | 24 | 106.96 | ||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
| Intercept | 4.799258185 | 1.197185164 | 4.008785216 | 0.00063622 | 2.309575344 | 7.288941027 |
| Hours of Unpaid Work per Week | -0.38184722 | 0.130750479 | -2.9204269 | 0.008177146 | -0.65375772 | -0.10993671 |
| Age | 0.004555815 | 0.022855709 | 0.199329423 | 0.843922453 | -0.04297523 | 0.052086864 |
| Income ([latex]\$1000[/latex]s) | 0.023250418 | 0.007610353 | 3.055103771 | 0.006012895 | 0.007423823 | 0.039077013 |
The [latex]p-\text{value}[/latex] for the test on the age regression coefficient is in the bottom part of the table under the P-value column of the Age row. So the [latex]p-\text{value}=0.8439[/latex].
Conclusion:
Because [latex]p-\text{value}=0.8439\gt 0.05=\alpha[/latex], we do not reject the null hypothesis. At the [latex]5\%[/latex] significance level, there is not enough evidence to suggest that there is a relationship between the dependent variable “job satisfaction” and the independent variable “age.”
NOTES
- The null hypothesis [latex]\beta_2=0[/latex] is the claim that the regression coefficient for the independent variable [latex]x_2[/latex] is zero. That is, the null hypothesis is the claim that there is no relationship between the dependent variable and the independent variable “age.”
- The alternative hypothesis is the claim that the regression coefficient for the independent variable [latex]x_2[/latex] is not zero. The alternative hypothesis is the claim that there is a relationship between the dependent variable and the independent variable “age.”
- When conducting a test on a regression coefficient, make sure to use the correct subscript on [latex]\beta[/latex] to correspond to how the independent variables were defined in the regression model and which independent variable is being tested. Here, the subscript on [latex]\beta[/latex] is [latex]2[/latex] because “age” is defined as [latex]x_2[/latex] in the regression model.
- The [latex]p-\text{value}[/latex] of [latex]0.8439[/latex] is a large probability compared to the significance level and so is likely to happen assuming the null hypothesis is true. This suggests that the assumption that the null hypothesis is true is most likely correct, and so the conclusion of the test is to not reject the null hypothesis. In other words, the regression coefficient [latex]\beta_2[/latex] is zero, and so there is no relationship between the dependent variable “job satisfaction” and the independent variable “age.” This means that the independent variable “age” is not particularly useful in predicting the dependent variable.
EXAMPLE
The human resources department at a large company wants to develop a model to predict an employee’s job satisfaction from the number of hours of unpaid work per week the employee does, the employee’s age, and the employee’s income. A sample of [latex]25[/latex] employees at the company is taken, and the data is recorded in the table below. The employee’s income is recorded in [latex]\$1000[/latex]s, and the job satisfaction score is out of [latex]10[/latex], with higher values indicating greater job satisfaction.
| Job Satisfaction | Hours of Unpaid Work per Week | Age | Income ([latex]\$1000[/latex]s) |
|---|---|---|---|
| 4 | 3 | 23 | 60 |
| 5 | 8 | 32 | 114 |
| 2 | 9 | 28 | 45 |
| 6 | 4 | 60 | 187 |
| 7 | 3 | 62 | 175 |
| 8 | 1 | 43 | 125 |
| 7 | 6 | 60 | 93 |
| 3 | 3 | 37 | 57 |
| 5 | 2 | 24 | 47 |
| 5 | 5 | 64 | 128 |
| 7 | 2 | 28 | 66 |
| 8 | 1 | 66 | 146 |
| 5 | 7 | 35 | 89 |
| 2 | 5 | 37 | 56 |
| 4 | 0 | 59 | 65 |
| 6 | 2 | 32 | 95 |
| 5 | 6 | 76 | 82 |
| 7 | 5 | 25 | 90 |
| 9 | 0 | 55 | 137 |
| 8 | 3 | 34 | 91 |
| 7 | 5 | 54 | 184 |
| 9 | 1 | 57 | 60 |
| 7 | 0 | 68 | 39 |
| 10 | 2 | 66 | 187 |
| 5 | 0 | 50 | 49 |
Previously, we found the multiple regression equation to predict the job satisfaction score from the other variables:
[latex]\begin{eqnarray*}\hat{y}&=&4.7993-0.3818x_1+0.0046x_2+0.0233x_3\\\\\hat{y}&=&\text{predicted job satisfaction score}\\x_1&=&\text{hours of unpaid work per week}\\x_2&=&\text{age}\\x_3&=&\text{income (\$1000s)}\end{eqnarray*}[/latex]
At the [latex]5\%[/latex] significance level, test the relationship between the dependent variable “job satisfaction” and the independent variable “income”.
Solution
Hypotheses:
[latex]\begin{eqnarray*}H_0:&&\beta_3=0\\H_a:&&\beta_3\neq 0\end{eqnarray*}[/latex]
[latex]p-\text{value}[/latex]:
The regression summary table generated by Excel is shown below:
| SUMMARY OUTPUT | ||||||
|---|---|---|---|---|---|---|
| Regression Statistics | ||||||
| Multiple R | 0.711779225 | |||||
| R Square | 0.506629665 | |||||
| Adjusted R Square | 0.436148189 | |||||
| Standard Error | 1.585212784 | |||||
| Observations | 25 | |||||
| ANOVA | ||||||
| >df | SS | MS | F | Significance F | ||
| Regression | 3 | 54.189109 | 18.06303633 | 7.18812504 | 0.001683189 | |
| Residual | 21 | 52.770891 | 2.512899571 | |||
| Total | 24 | 106.96 | ||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
| Intercept | 4.799258185 | 1.197185164 | 4.008785216 | 0.00063622 | 2.309575344 | 7.288941027 |
| Hours of Unpaid Work per Week | -0.38184722 | 0.130750479 | -2.9204269 | 0.008177146 | -0.65375772 | -0.10993671 |
| Age | 0.004555815 | 0.022855709 | 0.199329423 | 0.843922453 | -0.04297523 | 0.052086864 |
| Income ([latex]\$1000[/latex]s) | 0.023250418 | 0.007610353 | 3.055103771 | 0.006012895 | 0.007423823 | 0.039077013 |
The [latex]p-\text{value}[/latex] for the test on the income regression coefficient is in the bottom part of the table under the P-value column of the Income row. So the [latex]p-\text{value}=0.0060[/latex].
Conclusion:
Because [latex]p-\text{value}=0.0060\lt 0.05=\alpha[/latex], we reject the null hypothesis in favour of the alternative hypothesis. At the [latex]5\%[/latex] significance level, there is enough evidence to suggest that there is a relationship between the dependent variable “job satisfaction” and the independent variable “income.”
NOTES
- The null hypothesis [latex]\beta_3=0[/latex] is the claim that the regression coefficient for the independent variable [latex]x_3[/latex] is zero. That is, the null hypothesis is the claim that there is no relationship between the dependent variable and the independent variable “income.”
- The alternative hypothesis is the claim that the regression coefficient for the independent variable [latex]x_3[/latex] is not zero. The alternative hypothesis is the claim that there is a relationship between the dependent variable and the independent variable “income.”
- When conducting a test on a regression coefficient, make sure to use the correct subscript on [latex]\beta[/latex] to correspond to how the independent variables were defined in the regression model and which independent variable is being tested. Here, the subscript on [latex]\beta[/latex] is [latex]3[/latex] because “income” is defined as [latex]x_3[/latex] in the regression model.
- The [latex]p-\text{value}[/latex] of [latex]0.0060[/latex] is a small probability compared to the significance level and so is unlikely to happen assuming the null hypothesis is true. This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis. In other words, the regression coefficient [latex]\beta_3[/latex] is not zero, and so there is a relationship between the dependent variable “job satisfaction” and the independent variable “income.” This means that the independent variable “income” is useful in predicting the dependent variable.
Exercises
- A local restaurant advocacy group wants to study the relationship between a restaurant’s average weekly profit, the restaurant’s seating capacity, and the average daily traffic that passes the restaurant’s location. The group took a sample of restaurants and recorded their average weekly profit (in [latex]\$1000[/latex]s), the seating restaurant’s seating capacity, and the average number of cars (in [latex]1000[/latex]s) that passes the restaurant’s location. The data is recorded in the following table:
Seating Capacity Traffic Count ([latex]1000[/latex]s) Weekly Net Profit ([latex]\$1000[/latex]s) 120 19 23.8 180 8 29.2 150 12 22 180 15 26.2 220 16 33.5 235 10 32 115 18 22.4 110 12 20.4 165 21 23.7 220 20 34.7 140 24 27.1 145 24 23.3 140 13 20.9 200 14 29.6 210 14 31.4 175 12 23.2 175 15 31.1 190 17 28.2 100 23 25.2 145 20 20.7 135 13 37.2 25 13 26.3 140 25 20 130 14 28.2 135 10 24.6 160 23 23.7 In Question 1 of Section 13.1, we found the regression model to predict the average weekly profit from other variables.
- At the [latex]5\%[/latex] significance level, test the coefficient of seating capacity.
- At the [latex]5\%[/latex] significance level, test the coefficient of traffic count.
Click to see Answer
-
- Hypotheses: [latex]\begin{eqnarray*}H_0:&&\beta_1=0\\H_a:&&\beta_1\neq 0\end{eqnarray*}[/latex]
- [latex]p-\text{value}=0.0144[/latex]
- At the [latex]5\%[/latex] significance level, there is enough evidence to suggest that there is a relationship between the dependent variable “weekly profit” and the independent variable “seating capacity”.
- Hypotheses: [latex]\begin{eqnarray*}H_0:&&\beta_2=0\\H_a:&&\beta_2\neq 0\end{eqnarray*}[/latex]
- [latex]p-\text{value}=0.2645[/latex]
- At the [latex]5\%[/latex] significance level, there is not enough evidence to suggest that there is a relationship between the dependent variable “weekly profit” and the independent variable “traffic count”.
- A local university wants to study the relationship between a student’s GPA, the average number of hours they spend studying each night, and the average number of nights they go out each week. The university took a sample of students and recorded the following data:
GPA Average Number of Hours Spent Studying Each Night Average Number of Nights Go Out Each Week 3.72 5 1 3.88 3 1 3.67 2 1 3.87 3 4 2.49 1 4 1.29 1 2 1.01 2 4 2.12 1 1 1.9 1 5 3.42 3 2 1.33 1 4 1.07 0 2 2.75 3 1 3.82 4 1 3.91 5 0 2.25 2 3 2.06 1 5 2.92 3 2 3.06 3 1 3.65 2 2 3.69 4 1 In Question 2 of Section 13.1, we found the regression model to predict GPA from other variables.
- At the [latex]5\%[/latex] significance level, test the coefficient of average number of hours spent studying each night.
- At the [latex]5\%[/latex] significance level, test the coefficient of the average number of nights go out each week.
Click to see Answer
-
- Hypotheses: [latex]\begin{eqnarray*}H_0:&&\beta_1=0\\H_a:&&\beta_1\neq 0\end{eqnarray*}[/latex]
- [latex]p-\text{value}=0.0009[/latex]
- At the [latex]1\%[/latex] significance level, there is enough evidence to suggest that there is a relationship between the dependent variable “GPA” and the independent variable “average number of hours spent studying each night”.
- Hypotheses: [latex]\begin{eqnarray*}H_0:&&\beta_2=0\\H_a:&&\beta_2\neq 0\end{eqnarray*}[/latex]
- [latex]p-\text{value}=0.5083[/latex]
- At the [latex]1\%[/latex] significance level, there is not enough evidence to suggest that there is a relationship between the dependent variable “GPA” and the independent variable “average number of nights go out each week”.
- A very large company wants to study the relationship between the salaries of employees in management positions, their age, the number of years the employee spent in college, and the number of years the employee has been with the company. A sample of management employees is taken, and the datais recorded below:
Age Years of College Years with Company Salary ([latex]\$1000[/latex]s) 60 8 29 317.3 33 3 5 97.3 57 6 27 263.1 32 4 5 101.3 31 6 3 114.2 61 8 19 350.4 41 7 8 146.9 35 4 2 91.7 51 6 21 198.2 50 8 10 196.5 57 5 15 105.7 49 6 18 118.3 62 7 27 305.2 52 8 26 239.9 39 4 8 145.9 42 7 5 175.4 62 4 24 219.4 60 4 22 202.1 65 3 21 196.3 40 4 10 143.9 62 6 29 408.7 53 7 5 145.2 48 8 5 175.1 61 5 6 152.7 38 7 3 99.7 40 7 12 174.9 45 7 7 149.2 58 7 14 282.8 38 4 3 95.7 41 5 18 232.8 In Question 3 of Section 13.1, we found the regression model to predict salary from other variables.
- At the [latex]1\%[/latex] significance level, test the coefficient of age.
- At the [latex]1\%[/latex] significance level, test the coefficient of years of college.
- At the [latex]1\%[/latex] significance level, test the coefficient of years with the company.
Click to see Answer
-
- Hypotheses: [latex]\begin{eqnarray*}H_0:&&\beta_1=0\\H_a:&&\beta_1\neq 0\end{eqnarray*}[/latex]
- [latex]p-\text{value}=0.2383[/latex]
- At the [latex]1\%[/latex] significance level, there is not enough evidence to suggest that there is a relationship between the dependent variable “salary” and the independent variable “age”.
- Hypotheses: [latex]\begin{eqnarray*}H_0:&&\beta_2=0\\H_a:&&\beta_2\neq 0\end{eqnarray*}[/latex]
- [latex]p-\text{value}=0.097[/latex]
- At the [latex]1\%[/latex] significance level, there is enough evidence to suggest that there is a relationship between the dependent variable “salary” and the independent variable “years of college”.
- Hypotheses: [latex]\begin{eqnarray*}H_0:&&\beta_3=0\\H_a:&&\beta_3\neq 0\end{eqnarray*}[/latex]
- [latex]p-\text{value}=0.0005[/latex]
- At the [latex]1\%[/latex] significance level, there is enough evidence to suggest that there is a relationship between the dependent variable “salary” and the independent variable “years with company”.
“13.6 Testing the Regression Coefficients” and “13.8 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
