Questions: A researcher hopes to determine whether the number of hours a person jogs per week is related to the person's age. Age, x: 20, 47, 43, 21, 54 Hours, y: 6, 2.5, 2.5, 5, 1.5 Use a graphing calculator to create a line of best fit for the data. Round slope and y intercept to two decimal places, if necessary.

Solution Steps

To find the slope \( m \) of the line of best fit, we use the formula: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] where \( n \) is the number of data points.

Given data:

• \( x = [20, 47, 43, 21, 54] \)
• \( y = [6, 2.5, 2.5, 5, 1.5] \)

Calculate:

• \( \sum x = 185 \)
• \( \sum y = 17.5 \)
• \( \sum xy = 20 \times 6 + 47 \times 2.5 + 43 \times 2.5 + 21 \times 5 + 54 \times 1.5 = 487.5 \)
• \( \sum x^2 = 20^2 + 47^2 + 43^2 + 21^2 + 54^2 = 9363 \)
• \( n = 5 \)

Substitute into the formula: \[ m = \frac{5(487.5) - (185)(17.5)}{5(9363) - (185)^2} = \frac{2437.5 - 3237.5}{46815 - 34225} = \frac{-800}{12590} \approx -0.0636 \]

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

Substitute the values: \[ b = \frac{17.5 - (-0.0636)(185)}{5} = \frac{17.5 + 11.766}{5} = \frac{29.266}{5} \approx 5.85 \]

The equation of the line of best fit is: \[ y = mx + b \] Substitute the calculated values of \( m \) and \( b \): \[ y = -0.0636x + 5.85 \]

Final Answer