Question

a data set is given.
(a) draw a scatter diagram. comment on the type of relation that appears to exist between x and y.
(b) given that \\( \overline { x } = 3.8333, s _ { x } = 2.1370, \overline { y } = 4.0000, s _ { y } = 1.6840 \\), and \\( r = - 0.9337 \\), determine the least-squares regression line.
(c) graph the least-squares regression line on the scatter diagram drawn in part (a).
(a) choose the correc
there appears to be \\( \square \\) relationship.

Explanation:

Step1: Calculate the slope \( b_1 \)

The formula for the slope of the least - squares regression line is \( b_1=r\frac{s_y}{s_x} \).
Given \( r = - 0.9337\), \(s_x=2.1370\), \(s_y = 1.6840\)

$$b_1=-0.9337\times\frac{1.6840}{2.1370}$$

$$b_1=-0.9337\times0.788$$

$$b_1\approx - 0.736$$

Step2: Calculate the intercept \( b_0 \)

The formula for the intercept is \( b_0=\bar{y}-b_1\bar{x}\).
Given \(\bar{x}=3.8333\), \(\bar{y}=4.0000\), \(b_1=-0.736\)

$$b_0 = 4-(-0.736)\times3.8333$$

$$b_0=4 + 2.821$$

$$b_0\approx6.821$$

Answer:

The least - squares regression line is \(\hat{y}=-0.736x + 6.821\)