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 13 6 9 4 3 11 10 8 5 7 12
y 19.24 13.16 17.90 8.24 5.24 19.28 18.76 16.68 10.88 15.10 19.44
ŷ = □ + □x (round to two decimal places as needed.)

Explanation:

Step1: Calculate necessary sums

First, we need to calculate \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \).

Given \( x \)-values: \( 13, 6, 9, 4, 3, 11, 10, 8, 5, 7, 12 \)
\( \sum x = 13 + 6 + 9 + 4 + 3 + 11 + 10 + 8 + 5 + 7 + 12 = 88 \)

\( y \)-values: \( 19.24, 13.16, 17.90, 8.24, 5.24, 19.28, 18.76, 16.68, 10.88, 15.10, 19.44 \)
\( \sum y = 19.24 + 13.16 + 17.90 + 8.24 + 5.24 + 19.28 + 18.76 + 16.68 + 10.88 + 15.10 + 19.44 = 163.94 \)

Now calculate \( \sum xy \):
\( 13\times19.24 = 249.12 \)
\( 6\times13.16 = 78.96 \)
\( 9\times17.90 = 161.10 \)
\( 4\times8.24 = 32.96 \)
\( 3\times5.24 = 15.72 \)
\( 11\times19.28 = 212.08 \)
\( 10\times18.76 = 187.60 \)
\( 8\times16.68 = 133.44 \)
\( 5\times10.88 = 54.40 \)
\( 7\times15.10 = 105.70 \)
\( 12\times19.44 = 233.28 \)
\( \sum xy = 249.12 + 78.96 + 161.10 + 32.96 + 15.72 + 212.08 + 187.60 + 133.44 + 54.40 + 105.70 + 233.28 = 1464.36 \)

Calculate \( \sum x^2 \):
\( 13^2 = 169 \)
\( 6^2 = 36 \)
\( 9^2 = 81 \)
\( 4^2 = 16 \)
\( 3^2 = 9 \)
\( 11^2 = 121 \)
\( 10^2 = 100 \)
\( 8^2 = 64 \)
\( 5^2 = 25 \)
\( 7^2 = 49 \)
\( 12^2 = 144 \)
\( \sum x^2 = 169 + 36 + 81 + 16 + 9 + 121 + 100 + 64 + 25 + 49 + 144 = 814 \)

Step2: Calculate the slope \( b \)

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

Substitute the values:
\( n\sum xy = 11\times1464.36 = 16107.96 \)
\( \sum x \sum y = 88\times163.94 = 14426.72 \)
\( n\sum x^2 = 11\times814 = 8954 \)
\( (\sum x)^2 = 88^2 = 7744 \)

Now calculate \( b \):
\( b = \frac{16107.96 - 14426.72}{8954 - 7744} = \frac{1681.24}{1210} \approx 1.39 \)

Step3: Calculate the y-intercept \( a \)

The formula for the y-intercept \( a \) is:
\( a = \bar{y} - b\bar{x} \)
where \( \bar{x} = \frac{\sum x}{n} = \frac{88}{11} = 8 \)
\( \bar{y} = \frac{\sum y}{n} = \frac{163.94}{11} \approx 14.90 \)

Substitute \( b \approx 1.39 \), \( \bar{x} = 8 \), and \( \bar{y} \approx 14.90 \):
\( a = 14.90 - 1.39\times8 = 14.90 - 11.12 = 3.78 \)

Answer:

\( \hat{y} = 3.78 + 1.39x \)