Question

find the equation for the least squares regression line of the data described below.
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, she looked up the height (in meters), x, and the number of gym
members who had climbed it last year, y.
climbing gym statistics

• the mean of the x - values is 19.943.
• the mean of the y - values is 572.653.
• the sample standard deviation of the x - values is 5.830.
• the sample standard deviation of the y - values is 116.117.
• the correlation coefficient of the data set is - 0.991.

round your answers to the nearest thousandth.
y = □x + □

Explanation:

Step1: Calculate the slope ($b_1$)

The formula for the slope of the least - squares regression line is \(b_1 = r\frac{s_y}{s_x}\), where \(r\) is the correlation coefficient, \(s_y\) is the sample standard deviation of \(y\) - values, and \(s_x\) is the sample standard deviation of \(x\) - values.
Given \(r=- 0.991\), \(s_y = 116.117\), \(s_x=5.830\)

$$b_1=-0.991\times\frac{116.117}{5.830}$$

$$b_1=-0.991\times19.917$$

$$b_1\approx - 19.748$$

Step2: Calculate the y - intercept ($b_0$)

The formula for the y - intercept is \(b_0=\bar{y}-b_1\bar{x}\), where \(\bar{x}\) is the mean of \(x\) - values and \(\bar{y}\) is the mean of \(y\) - values.
Given \(\bar{x}=19.943\), \(\bar{y}=572.653\), \(b_1=-19.748\)

$$b_0 = 572.653-(-19.748)\times19.943$$

$$b_0=572.653 + 19.748\times19.943$$

$$b_0=572.653+393.810$$

$$b_0\approx966.463$$

Answer:

\(y=-19.748x + 966.463\)