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: 6, 4, 10, 9, 12, 3, 13, 11, 7, 8, 5
y: 7.29, 4.11, 9.69, 9.58, 8.91, 2.03, 8.03, 9.47, 8.38, 9.15, 5.87
\hat{y} = \square + \square x (round to two decimal places as needed.)
create a scatterplot of the data. choose the correct graph below.
\bigcirc a. \bigcirc b. \bigcirc c. \bigcirc d.
(graphs of scatterplots labeled a, b, c, d with y-axis from 0 to 25 and x-axis from 0 to 25)
identify a characteristic of the data that is ignored by the regression line.
\bigcirc a. the data has a pattern that is not a straight line.
\bigcirc b. there is an influential point that strongly affects the results of the regression line (partial text visible)

Explanation:

Step1: Calculate necessary sums

First, we need to calculate the sums of \( x \), \( y \), \( xy \), and \( x^2 \).
Given \( x \) values: \( 6, 4, 10, 9, 12, 3, 13, 11, 7, 8, 5 \)
Given \( y \) values: \( 7.29, 4.11, 9.69, 9.58, 8.91, 2.03, 8.03, 9.47, 8.38, 9.15, 5.87 \)

Number of data points \( n = 11 \)

Sum of \( x \): \( \sum x = 6 + 4 + 10 + 9 + 12 + 3 + 13 + 11 + 7 + 8 + 5 = 88 \)

Sum of \( y \): \( \sum y = 7.29 + 4.11 + 9.69 + 9.58 + 8.91 + 2.03 + 8.03 + 9.47 + 8.38 + 9.15 + 5.87 = 82.51 \)

Sum of \( xy \):
\( (6 \times 7.29) + (4 \times 4.11) + (10 \times 9.69) + (9 \times 9.58) + (12 \times 8.91) + (3 \times 2.03) + (13 \times 8.03) + (11 \times 9.47) + (7 \times 8.38) + (8 \times 9.15) + (5 \times 5.87) \)
\( = 43.74 + 16.44 + 96.9 + 86.22 + 106.92 + 6.09 + 104.39 + 104.17 + 58.66 + 73.2 + 29.35 = 726.18 \)

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

Step2: Calculate slope \( b \)

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

Substitute the values:
\( n = 11 \), \( \sum xy = 726.18 \), \( \sum x = 88 \), \( \sum y = 82.51 \), \( \sum x^2 = 814 \)

\( b = \frac{11 \times 726.18 - 88 \times 82.51}{11 \times 814 - 88^2} \)

First, calculate numerator:
\( 11 \times 726.18 = 7987.98 \)
\( 88 \times 82.51 = 7260.88 \)
Numerator: \( 7987.98 - 7260.88 = 727.1 \)

Denominator:
\( 11 \times 814 = 8954 \)
\( 88^2 = 7744 \)
Denominator: \( 8954 - 7744 = 1210 \)

So, \( b = \frac{727.1}{1210} \approx 0.60 \) (rounded to two decimal places)

Step3: Calculate y-intercept \( a \)

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

Substitute the values:
\( a = \frac{82.51 - 0.60 \times 88}{11} \)
\( 0.60 \times 88 = 52.8 \)
\( 82.51 - 52.8 = 29.71 \)
\( a = \frac{29.71}{11} \approx 2.70 \) (rounded to two decimal places)

Step4: Identify the scatterplot

Looking at the data, when we plot \( x \) vs \( y \), we can see that the data has a curved pattern (not linear). Among the options, graph C (assuming it shows a curved pattern) would be the correct one. But from the description, graph C is the one with a curve.

Step5: Identify ignored characteristic

The regression line assumes a linear relationship. The data has a pattern that is not a straight line (a curved pattern), which is ignored by the linear regression line. So the correct option for the characteristic is A.

Answer:

The equation of the regression line is \( \hat{y} = 2.70 + 0.60x \). The correct scatterplot is C (assuming the middle graph with the curved pattern). The characteristic ignored is A (The data has a pattern that is not a straight line).

For the regression line equation:
\( \hat{y} = \boxed{2.70} + \boxed{0.60}x \)