(a)
To find out the estimated multiple regression equation and model to predict the weight of a perch by two explanatory variables.
(a)
Answer to Problem 28.13AYK
The estimated multiple regression equationis
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 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:
(b)
To explain how much of the variation in the weight of perch is explained by the model in part (a).
(b)
Answer to Problem 28.13AYK
Only
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
(c)
To explain does the ANOVA table indicate that at least one of the explanatory variables is helpful in predicting the weight of perch.
(c)
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 explain do the individual t tests indicate that both
(d)
Answer to Problem 28.13AYK
The individual t tests indicate that both
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
(e)
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.
(e)
Answer to Problem 28.13AYK
The estimated multiple regression equationis
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:
In this,
(f)
To explain how much of the variation in the weight of perch is explained by the model in part (e).
(f)
Answer to Problem 28.13AYK
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
(g)
To explain does the ANOVA table indicate that at least one of the explanatory variables is helpful in predicting the weight of perch.
(g)
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 describe how the individual t statistics changed when the interaction term was added.
(h)
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.
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Chapter 28 Solutions
Practice of Statistics in the Life Sciences
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