Question

use the given data to find the equation of the regression line. examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.
x 11 9 13 10 10 13 7 4 11 8 5
y 7.81 6.44 12.67 7.4 7.61 8.89 5.8 5.44 8.03 6.23 5.92
find the equation of the regression line.
y^=□ + □x
(round the y-intercept two decimal places as needed. round the slope to three decimal places as needed.)

Explanation:

Step1: Calculate necessary sums

First, we need to find the sums of \( x \), \( y \), \( xy \), and \( x^2 \).
Given \( x \) values: \( 11, 9, 13, 10, 10, 13, 7, 4, 11, 8, 5 \)
Given \( y \) values: \( 7.81, 6.44, 12.67, 7.4, 7.61, 8.89, 5.8, 5.44, 8.03, 6.23, 5.92 \)

Number of data points \( n = 11 \)

Sum of \( x \): \( \sum x = 11 + 9 + 13 + 10 + 10 + 13 + 7 + 4 + 11 + 8 + 5 = 91 \)

Sum of \( y \): \( \sum y = 7.81 + 6.44 + 12.67 + 7.4 + 7.61 + 8.89 + 5.8 + 5.44 + 8.03 + 6.23 + 5.92 = 82.24 \)

Sum of \( xy \):
\( (11\times7.81)+(9\times6.44)+(13\times12.67)+(10\times7.4)+(10\times7.61)+(13\times8.89)+(7\times5.8)+(4\times5.44)+(11\times8.03)+(8\times6.23)+(5\times5.92) \)
\( = 85.91 + 57.96 + 164.71 + 74 + 76.1 + 115.57 + 40.6 + 21.76 + 88.33 + 49.84 + 29.6 \)
\( = 714.38 \)

Sum of \( x^2 \):
\( 11^2 + 9^2 + 13^2 + 10^2 + 10^2 + 13^2 + 7^2 + 4^2 + 11^2 + 8^2 + 5^2 \)
\( = 121 + 81 + 169 + 100 + 100 + 169 + 49 + 16 + 121 + 64 + 25 \)
\( = 915 \)

Step2: Calculate slope (\( m \)) and y-intercept (\( b \))

The formula for the slope \( m \) of the regression line is:
\( m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)

Substitute the values:
\( m = \frac{11\times714.38 - 91\times82.24}{11\times915 - 91^2} \)
First, calculate numerator: \( 11\times714.38 = 7858.18 \), \( 91\times82.24 = 7483.84 \), so numerator \( = 7858.18 - 7483.84 = 374.34 \)
Denominator: \( 11\times915 = 10065 \), \( 91^2 = 8281 \), so denominator \( = 10065 - 8281 = 1784 \)
\( m = \frac{374.34}{1784} \approx 0.2098 \approx 0.210 \) (rounded to three decimal places)

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

Substitute the values:
\( b = \frac{82.24 - 0.210\times91}{11} \)
\( 0.210\times91 = 19.11 \)
\( 82.24 - 19.11 = 63.13 \)
\( b = \frac{63.13}{11} \approx 5.74 \) (rounded to two decimal places)

Answer:

\(\hat{y} = 5.74 + 0.210x\)