Question

find the equation for the least squares regression line of the data described below. coach petersen 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 athlete’s height and 3000 - meter run time. coach petersen 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. height (x): 157, 158, 167, 170, 176; 3000 - meter time (y): 8.72, 8.60, 8.95, 8.03, 7.57. round your answers to the nearest thousandth. y = x +

Explanation:

Step1: Identify variables and data points

Let \( x \) be height (cm) and \( y \) be 3000 - meter time (min). The data points are: \((157, 8.72)\), \((158, 8.60)\), \((167, 8.95)\), \((170, 8.03)\), \((176, 7.57)\)

Step2: Calculate necessary sums

• \( n = 5 \) (number of data points)
• \( \sum x = 157 + 158 + 167 + 170 + 176 = 828 \)
• \( \sum y = 8.72 + 8.60 + 8.95 + 8.03 + 7.57 = 41.87 \)
• \( \sum xy = (157\times8.72)+(158\times8.60)+(167\times8.95)+(170\times8.03)+(176\times7.57) \)

\( = 1369.04+1358.8 + 1494.65+1365.1+1332.32 = 6919.91 \)

• \( \sum x^{2}=157^{2}+158^{2}+167^{2}+170^{2}+176^{2} \)

\( = 24649+24964 + 27889+28900+30976 = 137378 \)

Step3: Calculate slope (\( m \)) and intercept (\( b \)) of regression line \( y = mx + b \)

The formula for slope \( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} \)
Substitute values:
\( m=\frac{5\times6919.91 - 828\times41.87}{5\times137378-(828)^{2}} \)
First, calculate numerator: \( 5\times6919.91=34599.55 \), \( 828\times41.87 = 34668.36 \), numerator \( = 34599.55-34668.36=- 68.81 \)
Denominator: \( 5\times137378 = 686890 \), \( 828^{2}=685584 \), denominator \( = 686890 - 685584 = 1306 \)
\( m=\frac{-68.81}{1306}\approx - 0.0527 \)

The formula for intercept \( b=\frac{\sum y - m\sum x}{n} \)
Substitute values: \( \sum y = 41.87 \), \( m\approx - 0.0527 \), \( \sum x = 828 \), \( n = 5 \)
\( b=\frac{41.87-(-0.0527)\times828}{5}=\frac{41.87 + 43.6356}{5}=\frac{85.5056}{5}=17.10112 \)

Step4: Form the regression equation

The least squares regression line is \( y=-0.053x + 17.101 \) (rounded to nearest thousandth)

Answer:

\( y=-0.053x + 17.101 \)