Question

find the equation for the least squares regression line of the data described below.
coach duncan is responsible for recruiting male athletes to join the european masters track and field team. to improve his recruitment strategies, he wants to investigate the connection between an athletes height and 3000 - meter run time.
coach duncan has recorded the heights of the men on the track and field team (in centimeters), x, and their best 3000 - meter times (in minutes), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

Step1: Calculate the necessary sums

First, we need to find the following sums: \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), and the number of data points \( n = 6 \).

The data points are:
\( (x_1, y_1) = (161, 8.55) \)
\( (x_2, y_2) = (163, 8.81) \)
\( (x_3, y_3) = (165, 7.76) \)
\( (x_4, y_4) = (174, 7.38) \)
\( (x_5, y_5) = (176, 7.46) \)
\( (x_6, y_6) = (182, 7.33) \)

Calculating \( \sum x \):
\( \sum x = 161 + 163 + 165 + 174 + 176 + 182 = 1021 \)

Calculating \( \sum y \):
\( \sum y = 8.55 + 8.81 + 7.76 + 7.38 + 7.46 + 7.33 = 47.29 \)

Calculating \( \sum xy \):
\( 161 \times 8.55 = 1376.55 \)
\( 163 \times 8.81 = 1436.03 \)
\( 165 \times 7.76 = 1280.4 \)
\( 174 \times 7.38 = 1284.12 \)
\( 176 \times 7.46 = 1312.96 \)
\( 182 \times 7.33 = 1334.06 \)
\( \sum xy = 1376.55 + 1436.03 + 1280.4 + 1284.12 + 1312.96 + 1334.06 = 8024.12 \)

Calculating \( \sum x^2 \):
\( 161^2 = 25921 \)
\( 163^2 = 26569 \)
\( 165^2 = 27225 \)
\( 174^2 = 30276 \)
\( 176^2 = 30976 \)
\( 182^2 = 33124 \)
\( \sum x^2 = 25921 + 26569 + 27225 + 30276 + 30976 + 33124 = 174091 \)

Step2: Calculate the slope \( m \)

The formula for the slope \( m \) of the least squares regression line is:
\( m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)

Substituting the values:
\( n = 6 \), \( \sum xy = 8024.12 \), \( \sum x = 1021 \), \( \sum y = 47.29 \), \( \sum x^2 = 174091 \)

First, calculate the numerator:
\( n\sum xy - \sum x \sum y = 6 \times 8024.12 - 1021 \times 47.29 \)
\( = 48144.72 - 48283.09 \)
\( = -138.37 \)

Then, calculate the denominator:
\( n\sum x^2 - (\sum x)^2 = 6 \times 174091 - (1021)^2 \)
\( = 1044546 - 1042441 \)
\( = 2105 \)

Now, calculate \( m \):
\( m = \frac{-138.37}{2105} \approx -0.0657 \) (rounded to the nearest thousandth)

Step3: Calculate the y-intercept \( b \)

The formula for the y-intercept \( b \) is:
\( b = \frac{\sum y - m \sum x}{n} \)

Substituting the values:
\( \sum y = 47.29 \), \( m \approx -0.0657 \), \( \sum x = 1021 \), \( n = 6 \)

\( b = \frac{47.29 - (-0.0657) \times 1021}{6} \)
\( = \frac{47.29 + 67.08}{6} \)
\( = \frac{114.37}{6} \approx 19.062 \) (rounded to the nearest thousandth)

Answer:

\( y = -0.066x + 19.062 \) (Note: The slope was rounded to -0.066 as -0.0657 is approximately -0.066 when rounded to the nearest thousandth)