Chapter 3.2, Problem 53E

(a)

To determine

To make: a scatterplot for the given data

Answer to Problem 53E

  

Explanation of Solution

Given:

Percent return	74	66	81	52	73	62	52	45	62	46	60	46	38
New adults	5	6	8	11	12	15	16	17	18	18	19	20	20

Calculation:

We put percent return (the explanatory variable) on the horizontal axis and new adults (the response variable) on the vertical axis.

The completed scatterplot for the given data is shown below.

  

Conclusion:

Hence, scatterplot is drawn.

(b)

To determine

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

Answer to Problem 53E

The least square regression lie is $\widehat{Y}=31.9-0.304X$

Explanation of Solution

Calculation:

Using the MINITAB, we get the equation of the least-squares regression line is given below:

  

The least square regression lie is $\widehat{Y}=31.9-0.304X$

Observe the above scatter plot for the data with regression line added.

Conclusion:

Hence, the least square regression lie is $\widehat{Y}=31.9-0.304X$

(c)

To determine

To explain: the slope and y intercept of the regression line

Answer to Problem 53E

The slope b of above regression line (-0.304) is the rate at which the predicted response Y changes among the line as the explanatory variable X changes.

The Y intercept a of above regression line 31.9 is the predicted response Y when the explanatory variable $X=0$ .

Explanation of Solution

The equation of the least-squares regression line,

  $\widehat{Y}=31.9-0.304X$

Here the slope b of above regression line (-0.304) is the rate at which the predicted response Y changes among the line as the explanatory variable X changes. Specifically, b is the predicted changes Y when X increases by 1 unit. Here b is negative; this shows if percent return increases by 1 unit then new adult joins decreases by 0.304 units.

The Y intercept a of above regression line 31.9 is the predicted response Y when the explanatory variable $X=0$ . In the case of bird colonies, this shows if percent return of adult birds in a colony that return from previous year is 0, 31.9 new adults that join the colony.

(d)

To determine

To predict: the how many birds will join another colony of sparrow hawks, to which 60% of the adults from the previous year return.

Answer to Problem 53E

This shows that 13.66 birds will join another colony of sparrow hawks, to which 60% of the adults from the previous year return.

Explanation of Solution

Calculation:

The equation of the least-squares regression line,

  $\widehat{Y}=31.9-0.304X$

For $X=60\%$ , predicted value of Y is,

  $\begin{array}{l}\widehat{Y}=31.9-0.304X\\ =31.9-0.304\left(60\right)\\ =13.66\end{array}$

This shows that 13.66 birds will join another colony of sparrow hawks, to which 60% of the adults from the previous year return.

Conclusion:

Hence, this shows that 13.66 birds will join another colony of sparrow hawks, to which 60% of the adults from the previous year return.