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
four scatterplot images
find the equation of the regression line.
$hat{y} = \square + \square x$
(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 calculate the sum of \( x \) values (\( \sum x \)), sum of \( y \) values (\( \sum y \)), sum of \( x^2 \) (\( \sum x^2 \)), sum of \( xy \) (\( \sum xy \)), and the number of data points \( n \).

Given \( x \): 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

\( n = 11 \)

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

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

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

\( \sum xy = (11×7.81) + (9×6.44) + (13×12.67) + (10×7.4) + (10×7.61) + (13×8.89) + (7×5.8) + (4×5.44) + (11×8.03) + (8×6.23) + (5×5.92) \)
Calculating each term:
11×7.81 = 85.91
9×6.44 = 57.96
13×12.67 = 164.71
10×7.4 = 74
10×7.61 = 76.1
13×8.89 = 115.57
7×5.8 = 40.6
4×5.44 = 21.76
11×8.03 = 88.33
8×6.23 = 49.84
5×5.92 = 29.6

Summing these: \( 85.91 + 57.96 + 164.71 + 74 + 76.1 + 115.57 + 40.6 + 21.76 + 88.33 + 49.84 + 29.6 = 714.38 \)

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

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

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

First, calculate the numerator and denominator for \( m \):

Numerator for \( m \): \( 11×714.38 - 91×82.24 \)
\( = 7858.18 - 7483.84 = 374.34 \)

Denominator for \( m \): \( 11×915 - 91^2 \)
\( = 10065 - 8281 = 1784 \)

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

Now calculate \( b \):

\( b = \frac{82.24 - 0.2098×91}{11} \)
First, calculate \( 0.2098×91 \approx 19.0918 \)

Then, \( 82.24 - 19.0918 = 63.1482 \)

\( b = \frac{63.1482}{11} \approx 5.7407 \approx 5.74 \) (rounded to two decimal places)

Answer:

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