Chapter 3.2, Problem 63E

(a)

To determine

To find: the equation of the least-squares regression line

Answer to Problem 63E

The equation of the least-squares regression line is $\widehat{y}=-2.9935x+157.68$

There are $67.875\%$ of males that return after a season with 30 breeding pairs.

Explanation of Solution

Given:

  

Calculation:

From the above figure we can clearly see that $\begin{array}{l}\text{slope}\left(b\right)=-2.9935\\ Y-\text{intercept}\left(a\right)=157.68\end{array}$

Thus, the equation of the least-squares regression line is $\widehat{y}=-2.9935x+157.68$

The study records the percent of males (banded for identification) that returned the next breeding. This is the outcome of the study; the percent return is the response variable. Now, breeding pairs is a second variable. The entire purpose of the study is to determine how, if at all, breeding pairs affect the percent return. Breeding pairs is an explanatory variable. We will use breeding pairs (explanatory variable or predictor) to predict percent return (response or predicted variable). Percent return is the dependent variable and breeding pairs is the independent variable. The equation of the least-squares regression line for predicting the percent of males that return from the number of breeding pairs is.

Percent return $=157.68-2.9935$ (Breeding pairs)

The percent of returning males after a season with 30 breeding pairs is,$\begin{array}{c}\left(\text{Predicted}\right)\text{Percent return}=157.68-2.9935\times 30\\ =67.875\end{array}$

There are $67.875\%$ of males that return after a season with 30 breeding pairs.

Conclusion:

Therefore, the equation of the least-squares regression line is $\widehat{y}=-2.9935x+157.68$

There are $67.875\%$ of males that return after a season with 30 breeding pairs.

(b)

To determine

To find: percentage of variation explained by the regression line

Answer to Problem 63E

The percentage of variation explained by the regression line is $r^2=63.1\%$

Explanation of Solution

Calculation:

From the above figure we can clearly see that $r^2=63.1\%$

We interpret this by saying that $63.1\%$ of the year-to-year variation in percent of returning males is explained by the straight line relationship with number of breeding pairs the previous year.

Conclusion:

Therefore, the percentage of variation explained by the regression line is $r^2=63.1\%$

(c)

To determine

To find: the correlation of percent of returning males and breeding pairs and explain whether the sign of r is + or $-$ ?

Answer to Problem 63E

Percent of returning males and breeding pairs are negatively correlated and the coefficient of correlation between them is $-0.794$ .

Explanation of Solution

Calculation:

From the above figure we can clearly see that $\text{slope}\left(b\right)=-2.9935$

This shows negative correlation between variables.

  $\begin{array}{c}R-sq\left(R^2\right)=63.1\%\\ r^2=0.631\\ \text{correlation}\left(r\right)=\sqrt{r^2}\\ =\sqrt{0.631}\\ =-0.794\end{array}$

Percent of returning males and breeding pairs are negatively correlated and the coefficient of correlation between them is $-0.794$ .

Conclusion:

Therefore, percent of returning males and breeding pairs are negatively correlated and the coefficient of correlation between them is $-0.794$ .

(d)

To determine

To interpret: the value of s

Answer to Problem 63E

The predication of the percent returning is varied by about $9.46334$ % from the actual percent returning.

Explanation of Solution

Calculation:

For these data the standard deviation of the residuals:

  $s=9.46334$

The standard deviation of the residuals s measures the average size of the prediction errors (residuals) when using the regression line, $s=9.46334$ shows too much prediction error.

Conclusion: Therefore, the predication of the percent returning is varied by about $9.46334$ % from the actual percent returning.