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. height (in meters): 12.27, 13.13, 19.30, 19.43, 19.98; gym members: 812, 914, 508, 632, 535. round your answers to the nearest thousandth.

Explanation:

Step1: Identify variables and data points

Let \( x \) be the height (in meters) and \( y \) be the number of gym members. The data points are:
\((12.27, 812)\), \((13.13, 914)\), \((19.30, 508)\), \((19.43, 632)\), \((19.98, 535)\)

Step2: Calculate necessary sums

First, compute \( n = 5 \) (number of data points).
Calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \):

• \( \sum x = 12.27 + 13.13 + 19.30 + 19.43 + 19.98 = 84.11 \)
• \( \sum y = 812 + 914 + 508 + 632 + 535 = 3401 \)
• \( \sum xy = (12.27 \times 812) + (13.13 \times 914) + (19.30 \times 508) + (19.43 \times 632) + (19.98 \times 535) \)

\( = 9963.24 + 11990.82 + 9804.4 + 12289.76 + 10689.3 = 55737.52 \)

• \( \sum x^2 = (12.27)^2 + (13.13)^2 + (19.30)^2 + (19.43)^2 + (19.98)^2 \)

\( = 150.5529 + 172.3969 + 372.49 + 377.5249 + 399.2004 = 1472.1651 \)

Step3: Calculate slope (\( m \)) and intercept (\( b \))

The formula for the slope \( m \) of the least squares regression line \( y = mx + b \) is:
\[ m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \]
Substitute the values:
\[ m = \frac{5 \times 55737.52 - 84.11 \times 3401}{5 \times 1472.1651 - (84.11)^2} \]
First, compute numerator:
\( 5 \times 55737.52 = 278687.6 \)
\( 84.11 \times 3401 = 286058.11 \)
Numerator: \( 278687.6 - 286058.11 = -7370.51 \)

Denominator:
\( 5 \times 1472.1651 = 7360.8255 \)
\( (84.11)^2 = 7074.4921 \)
Denominator: \( 7360.8255 - 7074.4921 = 286.3334 \)

Thus, \( m = \frac{-7370.51}{286.3334} \approx -25.74 \) (rounded to nearest thousandth)

The formula for the intercept \( b \) is:
\[ b = \frac{\sum y - m \sum x}{n} \]
Substitute \( m \approx -25.74 \), \( \sum y = 3401 \), \( \sum x = 84.11 \), \( n = 5 \):
\[ b = \frac{3401 - (-25.74)(84.11)}{5} \]
Calculate \( (-25.74)(84.11) \approx -2165.0 \) (approximate for simplicity, but use precise \( m \))
Precise calculation: \( -25.74 \times 84.11 \approx -25.74 \times 84 + (-25.74) \times 0.11 = -2162.16 - 2.8314 = -2164.9914 \)
Thus, \( 3401 - (-2164.9914) = 3401 + 2164.9914 = 5565.9914 \)
\( b = \frac{5565.9914}{5} \approx 1113.20 \) (rounded to nearest thousandth)

Answer:

The least squares regression line is \( y = -25.74x + 1113.20 \) (rounded to the nearest thousandth for coefficients).

(Note: Depending on rounding precision during intermediate steps, slight variations may occur. For example, if \( m \) is calculated more precisely:
\( m = \frac{-7370.51}{286.3334} \approx -25.740 \), \( b = \frac{3401 - (-25.740)(84.11)}{5} \approx \frac{3401 + 2165.0}{5} = \frac{5566}{5} = 1113.200 \).)