Chapter 13, Problem 45CE

(a)

To determine

Find the regression model.

Answer to Problem 45CE

The regression model is $\text{Cost Per Load}=26.0000-6.3000\text{Topload}-0.2714\text{Powder}$

Explanation of Solution

Calculation:

The given information is that, a regression model is fitted to predict the cost per load for 20 types of laundry detergents using the binary predictors X₁ = top load (1 if washer is a top-loading model, 0 otherwise) and X₂ = Powder (if detergent was in powder form, 0 otherwise).

Multiple Regression equation:

The multiple regression equation with response variable Y which is related to k predictors $\left(X_1,X_2,\mathrm{...},X_k\right)$ is,

$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\mathrm{...}+{\beta }_k{x}_k+\epsilon$

In the equation, ${\beta }_0,{\beta }_1,\mathrm{...},{\beta }_k$ denotes the parameters that are unknown coefficients, $\epsilon$ denotes the random error.

The estimated multiple regression equation with response variable Y which is related to k predictors $\left(X_1,X_2,\mathrm{...},X_k\right)$ is,

$\widehat{y}=b_0+b_1{x}_1+b_2{x}_2+\mathrm{...}+b_k{x}_k$

In the equation, $\widehat{y}$ denotes the predicted value of response variable Y, $b_0,b_1,\mathrm{...},b_k$ denotes the estimated coefficients from sample.

The results have given that, the value of intercept is $b_0=26.0000$, the estimated coefficient for top load is $b_1=–6.3000$, for powder is $b_2=-0.2714$.

The fitted regression equation is,

$\text{Cost Per Load}=26.0000-6.3000\text{Topload}-0.2714\text{Powder}$.

(b)

To determine

State the conclusion about the overall fit of the model referring F statistic.

State the conclusion about the overall fit of the model referring p-value.

Answer to Problem 45CE

The conclusion about the overall fit of the model referring F statistic is all the coefficients are equal to zero and the overall regression is not significant.

The conclusion about the overall fit of the model referring p-value is all the coefficients are equal to zero and the overall regression is not significant.

Explanation of Solution

Calculation:

From the reported results, the value of F statistic is 1.06, the degrees of freedom for regression are 2, degrees of freedom for residual are 13 and the p-value is 0.3710. The considered level of significance is $\alpha =0.05$.

The formula for F statistic is,

$F_{\text{calc}}=\frac{MSR}{MSE}$

In the formula, MSR denotes the mean square regression and MSE denotes the mean square error.

Rejection rules based on F statistic:

• • If the test statistic value is greater than the critical value, then reject the null hypothesis. The regression is significant.
• • If the test statistic value is smaller than the critical value, then retain the null hypothesis. The regression is not significant.

Rejection rules based on p-value:

• • If p-value is less than the level of significance then the null hypothesis is rejected. The predictor is significant.
• • If p-value is greater than the level of significance then the null hypothesis is not rejected. The predictor is not significant.

Let ${\beta }_1$ denote the coefficient for predictor X₁ = top load (1 if washer is a top-loading model, 0 otherwise) and let ${\beta }_2$ denote the coefficient for predictor  X₂ = Powder (if detergent was in powder form, 0 otherwise).

Null hypothesis:

$H_0:$ All the coefficients are equal to zero $\left({\beta }_1={\beta }_2=0\right)$.

Alternative hypothesis:

$H_1:$ At least one of the coefficients is not zero.

Critical value:

The considered significance level is $\alpha =0.05$.

The degrees of freedom for numerator are 2, the degrees of freedom for denominator are 16 from completed F table.

From the Appendix F: Critical values of $F_{.05}$:

• • Locate the value 2 in numerator degrees of freedom $\left(d{f}_1\right)$ row.
• • Locate the value 16 in denominator degrees of freedom $\left(d{f}_2\right)$ column.
• • The intersecting value that corresponds to the (2, 16) with level of significance 0.05 is 3.63.

Hence, the critical value for $df=\left(2,16\right)$ with 0.05, level of significance is 3.63.

Conclusion referring F statistic:

The value of test statistic is 1.06.

The critical value is 3.63.

The test statistic value is less than the critical value.

The null hypothesis is not rejected.

The decision is that all the coefficients are equal to zero and the overall regression is not significant.

Conclusion referring p-value:

The p-value for overall regression is 0.3710.

The level of significance is 0.05.

The p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.3710\right)>\alpha \left(=0.05\right)$.

The null hypothesis is not rejected.

The decision is that all the coefficients are equal to zero and the overall regression is not significant.

(c)

To determine

State the conclusion for each individual predictor’s significance.

Answer to Problem 45CE

The predictor top-load is not significant. The predictor variable top-load is not related to cost per load.

The predictor powder is not significant. The predictor variable powder is not related to cost per load.

Explanation of Solution

Calculation:

From the reported results, the p-value for predictor Top-load is 0.1881 and p-value for predictor powder is 0.9270. The level of significance considered is $\alpha =0.05$.

For top-load:

Let ${\beta }_1$ is the parameter for the predictor top-load.

Null hypothesis:

$H_0:{\beta }_1=0$

The predictor variable top-load is not related to cost per load.

Alternative hypothesis:

$H_1:{\beta }_1\ne 0$

The predictor variable top-load is related to cost per load.

Conclusion:

The p-value for predictor top load is 0.1881.

The level of significance is 0.05.

The p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.1881\right)>\alpha \left(=0.05\right)$.

The null hypothesis is not rejected.

The predictor variable top-load is not related to cost per load.

The predictor top-load is not significant.

For seats-Patio:

Let ${\beta }_2$ is the parameter for the predictor powder.

Null hypothesis:

$H_0:{\beta }_2=0$

The predictor variable powder is not related to cost per load.

Alternative hypothesis:

$H_1:{\beta }_2\ne 0$

The predictor variable powder is related to cost per load.

Conclusion:

The p-value for predictor powder is 0.9270.

The level of significance is 0.05.

The p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.9270\right)>\alpha \left(=0.05\right)$.

The null hypothesis is not rejected.

The predictor variable powder is not related to cost per load.

The predictor powder is not significant.