Question

Use the least squares regression line of this data set to predict a value.
A climbing gym franchise is planning to install a new climbing wall at one of its locations. To help decide on the optimal height for the new wall, the location manager looked at usage patterns for the existing climbing walls.
For each wall, he looked up the height (in meters), x, and the number of gym members who had climbed it last year, y.
Height (in meters) Gym members
5.96 999
7.15 918
9.45 723
14.03 748
18.40 731
The least squares regression line of this data set is:
y=
–
18.822x+1,030.806
How many climbers does this line predict for a 13.91 meter wall?
Round your answer to the nearest integer.
gym members

Answer 1

To predict the number of gym members who would climb a wall that is 13.91 meters tall using the given least squares regression line equation:

\[ y = -18.822x + 1030.806 \]

we substitute \( x = 13.91 \) into the equation:

\[
y = -18.822(13.91) + 1030.806
\]

Calculating \( -18.822 \times 13.91 \):

\[
-18.822 \times 13.91 \approx -261.644
\]

Now substituting back into the equation:

\[
y \approx -261.644 + 1030.806
\]

Calculating the sum:

\[
y \approx 769.162
\]

Rounding to the nearest integer:

\[
y \approx 769
\]

Thus, the least squares regression line predicts that approximately **769 gym members** would climb the 13.91 meter wall.