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.
ŷ = □ + □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 \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \) for the given data.
The data points are:
\( x: 11, 9, 13, 10, 10, 13, 7, 4, 11, 8, 5 \)
\( y: 7.81, 6.44, 12.67, 7.4, 7.61, 8.89, 5.8, 5.44, 8.03, 6.23, 5.92 \)

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

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

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

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

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

The formula for the slope \( m \) of the regression line is:
\( m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)
where \( n = 11 \) (number of data points)

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

First, calculate \( m \):
\( n\sum xy = 11\times714.38 = 7858.18 \)
\( \sum x \sum y = 91\times82.24 = 7483.84 \)
\( n\sum x^2 = 11\times915 = 10065 \)
\( (\sum x)^2 = 91^2 = 8281 \)

So,
\( m = \frac{7858.18 - 7483.84}{10065 - 8281} = \frac{374.34}{1784} \approx 0.2098 \approx 0.210 \) (rounded to three decimal places)

Now, calculate \( b \):
\( m\sum x = 0.2098\times91 \approx 19.0918 \)
\( \sum y - m\sum x = 82.24 - 19.0918 = 63.1482 \)
\( b = \frac{63.1482}{11} \approx 5.74 \) (rounded to two decimal places)

Answer:

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