Question

use the given data to find the equation of the regression line. examine the scatterplot and identify a characteristic the data that is ignored by the regression line.
x 9 4 7 10 6 13 5 11 8 14 12
y 8.81 1.55 7.04 9.11 5.60 7.76 3.76 9.04 8.11 6.55 8.60
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 \( \sum x^2 \) for the given data.
The data points are:
\( x: 9, 4, 7, 10, 6, 13, 5, 11, 8, 14, 12 \)
\( y: 8.81, 1.55, 7.04, 9.11, 5.60, 7.76, 3.76, 9.04, 8.11, 6.55, 8.60 \)

Calculating \( \sum x \):
\( \sum x = 9 + 4 + 7 + 10 + 6 + 13 + 5 + 11 + 8 + 14 + 12 = 99 \)

Calculating \( \sum y \):
\( \sum y = 8.81 + 1.55 + 7.04 + 9.11 + 5.60 + 7.76 + 3.76 + 9.04 + 8.11 + 6.55 + 8.60 = 76.93 \)

Calculating \( \sum xy \):
\( (9\times8.81) + (4\times1.55) + (7\times7.04) + (10\times9.11) + (6\times5.60) + (13\times7.76) + (5\times3.76) + (11\times9.04) + (8\times8.11) + (14\times6.55) + (12\times8.60) \)
\( = 79.29 + 6.2 + 49.28 + 91.1 + 33.6 + 100.88 + 18.8 + 99.44 + 64.88 + 91.7 + 103.2 \)
\( = 748.37 \)

Calculating \( \sum x^2 \):
\( 9^2 + 4^2 + 7^2 + 10^2 + 6^2 + 13^2 + 5^2 + 11^2 + 8^2 + 14^2 + 12^2 \)
\( = 81 + 16 + 49 + 100 + 36 + 169 + 25 + 121 + 64 + 196 + 144 \)
\( = 949 \)

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)

Substituting the values:
\( n\sum xy = 11\times748.37 = 8232.07 \)
\( \sum x \sum y = 99\times76.93 = 7616.07 \)
\( n\sum x^2 = 11\times949 = 10439 \)
\( (\sum x)^2 = 99^2 = 9801 \)

\( b = \frac{8232.07 - 7616.07}{10439 - 9801} = \frac{616}{638} \approx 0.9655 \approx 0.97 \) (rounded to two decimal places)

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

The formula for the y-intercept \( a \) is:
\( a = \bar{y} - b\bar{x} \)
where \( \bar{x} = \frac{\sum x}{n} = \frac{99}{11} = 9 \)
\( \bar{y} = \frac{\sum y}{n} = \frac{76.93}{11} \approx 6.9936 \)

Substituting the values:
\( a = 6.9936 - 0.9655\times9 \)
\( = 6.9936 - 8.6895 \)
\( = -1.6959 \approx -1.70 \) (rounded to two decimal places)

Answer:

\( \hat{y} = -1.70 + 0.97x \)