Chapter 12: Q. 29 (page 804)

Park rangers are interested in estimating the weight
of the bears that inhabit their state. The rangers have data
on weight (in pounds) and neck girth (distance around
the neck in inches) for 10 randomly selected bears. Some
regression output for these data is shown below.
A bear was recently captured whose neck girth was $35 i n c h e s$ and whose weight was $466.35 p o u n d s$ . If this bear were added to the data set given above, what would be the effect on the value of $r_{2}$ ?
(a) It would decrease the value of $r_{2}$ because the added
point is an outlier.
(b) It would increase the value of $r_{2}$ because any point
added to the data would increase the percent of variation
in bear weight that can be explained by the least-squares
regression line.
(c) It would increase the value of $r_{2}$ because the added
point lies on the least-squares regression line and is far from
the point $(x, y)$
(d) It would have no effect on the value of $r_{2}$ because the
added point lies far from the point $(x, y)$
(e) It would have no effect on the value of $r_{2}$ because it lies
on the least-squares regression line.

Short Answer

Expert verified

The option (b) It would increase the value of $r_{2}$ because any point added to the data would increase the percent of variation in bear weight that can be explained by the least-squares regression line is correct .

Step by step solution

Given information

We are given that A bear was recently captured whose neck girth was $35 i n c h e s$ and whose weight was $466.35 p o u n d s$ . If this bear were added to the data set given above, what would be the effect on the value of $r_{2}$ ?

we need to find that what would be the effect on the value of $r_{2}$ ?

Explanation

We note that if we would add the bear with neck girth $35 i n c h e s$ and weight 466.35 pounds to the scatterplot . moreover , this plot will lie in the extension of Least square regression line and thus will lie on least square regression line.

Moreover, standard error of estimation s will then also decrease ,because the Added data will have a prefer prediction . Thus our predictions are more accurate Resulting in less variations and thus a lower standard error of the estimation s.

so option (b) It would increase the value of $r_{2}$ because any point added to the data would increase the percent of variation in bear weight that can be explained by the least-squares regression line is correct .