Question

listed below are numbers of internet users per 100 people and numbers of scientific award winners per 10 million people for different countries. construct a scatterplot, find the linear correlation coefficient r, and find the p - value of r. determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. use significance level of α = 0.01.
internet users | 79.9 80.4 57.7 66.4 78.3 38.4
award winners | 5.3 9.1 3.3 1.8 10.5 0.1
construct a scatterplot. choose the correct graph below
a.
b.
c.
d.
the linear correlation coefficient is r = \boxed{}.
(round to three decimal places as needed.)

Explanation:

Step1: Identify Variables

Let \( x \) be Internet Users (per 100 people): \( [79.9, 80.4, 57.7, 66.4, 78.3, 38.4] \)
Let \( y \) be Award Winners (per 10 million people): \( [5.3, 9.1, 3.3, 1.8, 10.5, 0.1] \)

Step2: Calculate Necessary Sums

• \( n = 6 \)
• \( \sum x = 79.9 + 80.4 + 57.7 + 66.4 + 78.3 + 38.4 = 401.1 \)
• \( \sum y = 5.3 + 9.1 + 3.3 + 1.8 + 10.5 + 0.1 = 30.1 \)
• \( \sum xy = (79.9×5.3) + (80.4×9.1) + (57.7×3.3) + (66.4×1.8) + (78.3×10.5) + (38.4×0.1) \)

\( = 423.47 + 731.64 + 190.41 + 119.52 + 822.15 + 3.84 = 2291.03 \)

• \( \sum x^2 = 79.9^2 + 80.4^2 + 57.7^2 + 66.4^2 + 78.3^2 + 38.4^2 \)

\( = 6384.01 + 6464.16 + 3329.29 + 4408.96 + 6130.89 + 1474.56 = 28191.87 \)

• \( \sum y^2 = 5.3^2 + 9.1^2 + 3.3^2 + 1.8^2 + 10.5^2 + 0.1^2 \)

\( = 28.09 + 82.81 + 10.89 + 3.24 + 110.25 + 0.01 = 235.29 \)

Step3: Apply Correlation Formula

The formula for \( r \) 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 values:

• Numerator: \( 6×2291.03 - 401.1×30.1 = 13746.18 - 12073.11 = 1673.07 \)
• Denominator part 1: \( 6×28191.87 - (401.1)^2 = 169151.22 - 160881.21 = 8270.01 \)
• Denominator part 2: \( 6×235.29 - (30.1)^2 = 1411.74 - 906.01 = 505.73 \)
• Denominator: \( \sqrt{8270.01×505.73} \approx \sqrt{4182490.6573} \approx 2045.11 \)
• \( r = \frac{1673.07}{2045.11} \approx 0.818 \)

Answer:

\( 0.818 \)