Chapter 28, Problem 28.13AYK

(a)

To determine

To find out the estimated multiple regression equation and model to predict the weight of a perch by two explanatory variables.

Answer to Problem 28.13AYK

The estimated multiple regression equationis $\widehat{y}=-578.758+14.307x_1+113.50x_2$ .

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. We will use the Excel software to find the regression analysis by the data analysis option in the data tab. Now, the estimated regression model with two explanatory variables, length and width, to predict perch is as:

Regression Statistics
Multiple R	0.968139
R Square	0.937292
Adjusted R Square	0.934926
Standard Error	88.676
Observations	56

ANOVA
	df	SS	MS	F	Significance F
Regression	2	6229332	3114666	396.095	1.35E-32
Residual	53	416761.9	7863.433
Total	55	6646094

	Coefficients	Standard Error	t Stat	P-value
Intercept	-578.758	43.66725	-13.2538	1.85E-18
Length	14.30738	5.658797	2.528344	0.014475
Width	113.4997	30.26474	3.750227	0.000439

Thus, from above the multiple regression equation is as:

  $\begin{array}{l}\widehat{y}=b_0+b_1{x}_1+b_2{x}_2\\ \Rightarrow \widehat{y}=-578.758+14.307x_1+113.50x_2\end{array}$

(b)

To determine

To explain how much of the variation in the weight of perch is explained by the model in part (a).

Answer to Problem 28.13AYK

Only $93.73\%$ of the variation in the weight of perch is explained by the model in part (a).

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. We will use the Excel software to find the regression analysis by the data analysis option in the data tab. Now, the estimated regression model with two explanatory variables, length and width, to predict perch is as:

Regression Statistics
Multiple R	0.968139
R Square	0.937292
Adjusted R Square	0.934926
Standard Error	88.676
Observations	56

As we can see that $R^2=93.73\%$ that is the coefficient of determination and this explains the variation in the weight of perch is explained by the model in part (a).

(c)

To determine

To explain does the ANOVA table indicate that at least one of the explanatory variables is helpful in predicting the weight of perch.

Answer to Problem 28.13AYK

The ANOVA table indicates that at least one of the explanatory variables is helpful in predicting the weight of perch.

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. From part (a), we can say that the test statistics value of F is more than the critical value of F , then it means that we can reject the null hypothesis and there is statistical significance and this explains that the ANOVA table indicates that at least one of the explanatory variables is helpful in predicting the weight of perch.

(d)

To determine

To explain do the individual t tests indicate that both ${\beta }_1\text{ and }{\beta }_2$ are significantly different from zero.

Answer to Problem 28.13AYK

The individual t tests indicate that both ${\beta }_1\text{ and }{\beta }_2$ are significantly different from zero.

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. From part (a), we can say that the P-values of both the ${\beta }_1\text{ and }{\beta }_2$ are significantly smaller than the level of significance then it means that we can reject the null hypothesis and there is statistical significance and that implies the individual t tests indicate that both ${\beta }_1\text{ and }{\beta }_2$ are significantly different from zero.

(e)

To determine

To create a new variable called interaction and use the multiple regression model with three explanatory variables to predict weight of a perch and the estimated multiple regression equation.

Answer to Problem 28.13AYK

The estimated multiple regression equationis $\widehat{y}=113.935-3.483x_1-94.631x_2+5.241x_1{x}_2$ .

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. Now, we will create a new variable called interaction by multiplying the two explanatory variables, length and width and use the Excel software to find the regression analysis by the data analysis option in the data tab. Now, the estimated regression model with three explanatory variables, length and width, to predict perch is as:

Regression Statistics
Multiple R	0.992314
R Square	0.984688
Adjusted R Square	0.983805
Standard Error	44.23814
Observations	56

ANOVA
	df	SS	MS	F	Significance F
Regression	3	6544330	2181443	1114.68	3.75E-47
Residual	52	101764.7	1957.013
Total	55	6646094

	Coefficients	Standard Error	t Stat	P-value
Intercept	113.9349	58.78439	1.938183	0.058039
Length	-3.48269	3.152101	-1.10488	0.274298
Width	-94.6309	22.29543	-4.24441	9.06E-05
interaction	5.241238	0.413121	12.68693	1.52E-17

And the estimated multiple regression equation with the term interaction is as:

  $\begin{array}{l}\widehat{y}=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_1{x}_2\\ \Rightarrow \widehat{y}=113.935-3.483x_1-94.631x_2+5.241x_1{x}_2\end{array}$

In this, $x_1{x}_2$ refer to the term interaction of both the variables.

(f)

To determine

To explain how much of the variation in the weight of perch is explained by the model in part (e).

Answer to Problem 28.13AYK

  $98.47\%$ of the variation in the weight of perch is explained by the model in part (e).

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. Now, we will create a new variable called interaction by multiplying the two explanatory variables, length and width and use the Excel software to find the regression analysis by the data analysis option in the data tab. Now, the estimated regression model with three explanatory variables, length and width, to predict perch is as:

Regression Statistics
Multiple R	0.992314
R Square	0.984688
Adjusted R Square	0.983805
Standard Error	44.23814
Observations	56

Since in this we have $R^2=98.47\%$ that is the coefficient of determination and this explains the variation in the weight of perch is explained by the model in part (e).

(g)

To determine

To explain does the ANOVA table indicate that at least one of the explanatory variables is helpful in predicting the weight of perch.

Answer to Problem 28.13AYK

Yes, the ANOVA table indicates that at least one of the explanatory variables is helpful in predicting the weight of perch.

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. From part (e), we can say from the analysis that the test statistics value of F is more than the critical value of F , then it means that we can reject the null hypothesis and there is statistical significance and this explains that the ANOVA table indicates that at least one of the explanatory variables is helpful in predicting the weight of perch.

(h)

To determine

To describe how the individual t statistics changed when the interaction term was added.

Explanation of Solution

In the question, it is given the table that contains data on the size of perch caught in a lake in Finland. From part (e) and (a), we can say from the analysis that when the interaction term is added to the model then the individual t statistics becomes negative and decreased from that of part (a) and also in part (e) one of the P-values for the slope is larger than the level of significance that means it is not statistically significant but in part (a), both the slopes were statistically significant.