Question

use the given data to find the equation of the regression line. round the final values to three significant digits, if necessary.

| x | 6 8 20 28 36
| y | 2 4 13 20 30

a. \\(\hat{y} = -2.79 + 0.897x\\)
b. \\(\hat{y} = -2.79 + 0.950x\\)
c. \\(\hat{y} = -3.79 + 0.801x\\)
d. \\(\hat{y} = -3.79 + 0.897x\\)

Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values: \( x = [6, 8, 20, 28, 36] \), so \( \bar{x} = \frac{6 + 8 + 20 + 28 + 36}{5} = \frac{98}{5} = 19.6 \).
Then, find the mean of \( y \) values: \( y = [2, 4, 13, 20, 30] \), so \( \bar{y} = \frac{2 + 4 + 13 + 20 + 30}{5} = \frac{69}{5} = 13.8 \).

Step2: Calculate slope (m) of regression line

The formula for slope \( m \) is \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \).

• Calculate \( (x_i - \bar{x})(y_i - \bar{y}) \) for each \( i \):
• \( (6 - 19.6)(2 - 13.8) = (-13.6)(-11.8) = 160.48 \)
• \( (8 - 19.6)(4 - 13.8) = (-11.6)(-9.8) = 113.68 \)
• \( (20 - 19.6)(13 - 13.8) = (0.4)(-0.8) = -0.32 \)
• \( (28 - 19.6)(20 - 13.8) = (8.4)(6.2) = 52.08 \)
• \( (36 - 19.6)(30 - 13.8) = (16.4)(16.2) = 265.68 \)

Sum: \( 160.48 + 113.68 - 0.32 + 52.08 + 265.68 = 591.6 \)

• Calculate \( (x_i - \bar{x})^2 \) for each \( i \):
• \( (6 - 19.6)^2 = (-13.6)^2 = 184.96 \)
• \( (8 - 19.6)^2 = (-11.6)^2 = 134.56 \)
• \( (20 - 19.6)^2 = (0.4)^2 = 0.16 \)
• \( (28 - 19.6)^2 = (8.4)^2 = 70.56 \)
• \( (36 - 19.6)^2 = (16.4)^2 = 268.96 \)

Sum: \( 184.96 + 134.56 + 0.16 + 70.56 + 268.96 = 659.2 \)

• Slope \( m = \frac{591.6}{659.2} \approx 0.897 \).

Step3: Calculate y-intercept (b)

The formula for y-intercept \( b = \bar{y} - m\bar{x} \).
Substitute \( \bar{y} = 13.8 \), \( m \approx 0.897 \), \( \bar{x} = 19.6 \):
\( b = 13.8 - (0.897)(19.6) \approx 13.8 - 17.5812 \approx -3.7812 \approx -3.79 \) (rounded to three significant digits).

Step4: Form regression equation

The regression line is \( \hat{y} = b + mx \), so \( \hat{y} = -3.79 + 0.897x \), which matches option D.

Answer:

D. \( \hat{y} = -3.79 + 0.897x \)