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 calculate the sums of \( x \), \( y \), \( xy \), and \( x^2 \) for the given data.
The data points are:
\( x: [11, 9, 13, 10, 10, 13, 7, 4, 11, 8, 5] \)
\( y: [7.81, 6.44, 12.67, 7.4, 7.61, 8.89, 5.8, 5.44, 8.03, 6.23, 5.92] \)

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 \( xy \) (\( \sum xy \)):
\( (11×7.81)+(9×6.44)+(13×12.67)+(10×7.4)+(10×7.61)+(13×8.89)+(7×5.8)+(4×5.44)+(11×8.03)+(8×6.23)+(5×5.92) \)
\( = 85.91 + 57.96 + 164.71 + 74 + 76.1 + 115.57 + 40.6 + 21.76 + 88.33 + 49.84 + 29.6 = 714.38 \)

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

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:
\( m = \frac{11×714.38 - 91×82.24}{11×915 - 91^2} \)
First, calculate numerator:
\( 11×714.38 = 7858.18 \)
\( 91×82.24 = 7483.84 \)
Numerator: \( 7858.18 - 7483.84 = 374.34 \)

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

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

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

Substitute the values:
\( b = \frac{82.24 - 0.2098×91}{11} \)
\( 0.2098×91 ≈ 19.0918 \)
\( 82.24 - 19.0918 = 63.1482 \)
\( b = \frac{63.1482}{11} ≈ 5.7407 ≈ 5.74 \) (rounded to two decimal places)

Answer:

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