Question

1. the equation of a least - squares regression line is \\( \hat { y } = - 0.157 x + 68.714 \\). here are the summarystatistics of the variables:\\( \overline { x } = 185.69 \\)\\( \overline { y } = 39.62 \\)\\( s _ { x } = 37.92 \\)\\( s _ { y } = 11.09 \\)what is the approximate correlation (r - value) of the least - squares regression line?\\( ( a ) - 0.157 \\)\\( ( b ) - 0.046 \\)\\( ( c ) - 0.536 \\)\\( ( d ) 0.287 \\)\\( ( e ) 20.1 \\)

Explanation:

Step1: Recall the formula for the slope of the regression line

The slope \( b_1 \) of the least - squares regression line is given by \( b_1=r\frac{s_y}{s_x} \), where \( r \) is the correlation coefficient, \( s_y \) is the standard deviation of the \( y \) - variable, and \( s_x \) is the standard deviation of the \( x \) - variable.
We can re - arrange the formula for \( r \) as \( r = b_1\frac{s_x}{s_y} \).

Step2: Identify the values from the problem

We are given that \( b_1=- 0.157 \), \( s_x = 37.92 \), and \( s_y=11.09 \).
Substitute these values into the formula for \( r \):
\( r=-0.157\times\frac{37.92}{11.09} \)
First, calculate \( \frac{37.92}{11.09}\approx3.42 \)
Then, \( r=-0.157\times3.42\approx - 0.537 \) (the small difference from - 0.536 is due to rounding differences in intermediate steps)

Answer:

C. - 0.536