Question

use the given data set to complete parts (a) through (c) below. (use α = 0.05.)

x	y
8	6.76
13	12.74
9	7.11
11	7.82
14	8.84
6	6.08
4	5.38
12	8.16
7	6.43
5	5.73

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

a. construct a scatterplot. choose the correct graph below.
options a, b, c, d (with scatterplot graphs)

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 = \boxed{ } (round to three decimal places as needed.)

Explanation:

Step1: List the data points

We have the following pairs of \( (x, y) \) values:
\( (10, 7.46), (8, 6.76), (13, 12.74), (9, 7.11), (11, 7.82), (14, 8.84), (6, 6.08), (4, 5.38), (12, 8.16), (7, 6.43), (5, 5.73) \)

Step2: Calculate necessary sums

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

• \( \sum x = 10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5 = 99 \)
• \( \sum y = 7.46 + 6.76 + 12.74 + 7.11 + 7.82 + 8.84 + 6.08 + 5.38 + 8.16 + 6.43 + 5.73 = 82.51 \)
• \( \sum xy = (10\times7.46)+(8\times6.76)+(13\times12.74)+(9\times7.11)+(11\times7.82)+(14\times8.84)+(6\times6.08)+(4\times5.38)+(12\times8.16)+(7\times6.43)+(5\times5.73) \)

\( = 74.6 + 54.08 + 165.62 + 63.99 + 86.02 + 123.76 + 36.48 + 21.52 + 97.92 + 45.01 + 28.65 = 797.65 \)

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

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

• \( \sum y^2 = 7.46^2 + 6.76^2 + 12.74^2 + 7.11^2 + 7.82^2 + 8.84^2 + 6.08^2 + 5.38^2 + 8.16^2 + 6.43^2 + 5.73^2 \)

\( \approx 55.65 + 45.69 + 162.31 + 50.55 + 61.15 + 78.15 + 36.97 + 28.94 + 66.59 + 41.34 + 32.83 = 660.17 \)

Step3: Use the formula for the correlation coefficient

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

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

where \( n = 11 \) (the number of data points).

First, calculate the numerator:
\( n\sum xy - (\sum x)(\sum y)=11\times797.65 - 99\times82.51 = 8774.15 - 8168.49 = 605.66 \)

Next, calculate the denominator:

• \( n\sum x^2 - (\sum x)^2 = 11\times1001 - 99^2 = 11011 - 9801 = 1210 \)
• \( n\sum y^2 - (\sum y)^2 = 11\times660.17 - 82.51^2 \approx 7261.87 - 6807.90 = 453.97 \)
• The product inside the square root: \( 1210\times453.97 \approx 549203.7 \)
• The square root: \( \sqrt{549203.7} \approx 741.1 \)

Now, calculate \( r \):
\( r=\frac{605.66}{741.1}\approx0.817 \)

Answer:

\( 0.817 \) (rounded to three decimal places)