Question

question 7, 10.1.11
part 2 of 4
hw score: 81.96%, 54.1 of 66 points
points: 1.25 of 5
use the given data set to complete parts (a) through (c) below. (use α = 0.05.)

10	8	13	9	11	14	6	4	12	7	5
9.15	8.15	8.74	8.78	9.27	8.09	6.12	3.09	9.13	7.27	4.74

click here to view a table of critical values for the correlation coefficient.

b. find the linear correlation coefficient, r, then determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.
the linear correlation coefficient is r =
(round to three decimal places as needed.)

Explanation:

Step1: Identify the data points

Let the \( x \)-values be \( [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5] \) and the \( y \)-values be \( [9.15, 8.15, 8.74, 8.78, 9.27, 8.09, 6.12, 3.09, 9.13, 7.27, 4.74] \). The number of data points \( n = 11 \).

Step2: Calculate necessary sums

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

• \( \sum x = 10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5 = 99 \)
• \( \sum y = 9.15 + 8.15 + 8.74 + 8.78 + 9.27 + 8.09 + 6.12 + 3.09 + 9.13 + 7.27 + 4.74 = 82.53 \)
• \( \sum xy = (10\times9.15)+(8\times8.15)+(13\times8.74)+(9\times8.78)+(11\times9.27)+(14\times8.09)+(6\times6.12)+(4\times3.09)+(12\times9.13)+(7\times7.27)+(5\times4.74) \)

\( = 91.5 + 65.2 + 113.62 + 79.02 + 101.97 + 113.26 + 36.72 + 12.36 + 109.56 + 50.89 + 23.7 = 797.8 \)

• \( \sum x^2 = 10^2 + 8^2 + 13^2 + 9^2 + 11^2 + 14^2 + 6^2 + 4^2 + 12^2 + 7^2 + 5^2 \)

\( = 100 + 64 + 169 + 81 + 121 + 196 + 36 + 16 + 144 + 49 + 25 = 1001 \)

• \( \sum y^2 = 9.15^2 + 8.15^2 + 8.74^2 + 8.78^2 + 9.27^2 + 8.09^2 + 6.12^2 + 3.09^2 + 9.13^2 + 7.27^2 + 4.74^2 \)

\( \approx 83.72 + 66.42 + 76.39 + 77.09 + 85.93 + 65.45 + 37.45 + 9.55 + 83.36 + 52.85 + 22.47 = 660.68 \)

Step3: Use the correlation coefficient formula

The formula for the linear correlation coefficient \( r \) is:

$$ r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

Substitute the values:

• Numerator: \( 11\times797.8 - 99\times82.53 = 8775.8 - 8170.47 = 605.33 \)
• Denominator part 1: \( 11\times1001 - 99^2 = 11011 - 9801 = 1210 \)
• Denominator part 2: \( 11\times660.68 - 82.53^2 \approx 7267.48 - 6810.20 = 457.28 \)
• Denominator: \( \sqrt{1210\times457.28} \approx \sqrt{553308.8} \approx 743.85 \)

Then, \( r = \frac{605.33}{743.85} \approx 0.814 \) (rounded to three decimal places)

Answer:

\( 0.814 \)