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)
\

$$\begin{tabular} { l l l l l l l } { x } & { 1 } & { 2 } & { 3 } & { 5 } & { 6 } & { 6 } \\hline { y } & { 5.2 } & { 5.9 } & { 5.4 } & { 3.0 } & { 2.1 } & { 2.4 } \\end{tabular}$$

there appears to be a linear, negative relationship.
(b) \\( \hat { y } = \square x + \square \\)
(round to thr decimal places as needed.)

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 \(r = - 0.9337\), \(s_y=1.6840\), \(s_x = 2.1370\)

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

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

$$b_1\approx - 0.736$$

Step2: Calculate the intercept \(b_0\)

The formula for the intercept of the least - squares regression line is \(b_0=\bar{y}-b_1\bar{x}\)
Given \(\bar{x}=3.8333\), \(\bar{y}=4.0000\), \(b_1=-0.736\)

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

$$b_0=4 + 0.736\times3.8333$$

$$b_0=4+2.821$$

$$b_0\approx6.821$$

Answer:

\(\hat{y}=-0.736x + 6.821\)