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 value of 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 a significance level of α = 0.05.
internet users | 78.8 80.2 56.3 67.2 76.7 38.0
award winners | 5.4 8.6 3.2 1.8 10.9 0.1
the linear correlation coefficient is r = \square.
(round to three decimal places as needed.)

Explanation:

Step1: Identify data points

Let \( x \) be Internet Users (\( x: 78.8, 80.2, 56.3, 67.2, 76.7, 38.0 \)) and \( y \) be Award Winners (\( y: 5.4, 8.6, 3.2, 1.8, 10.9, 0.1 \)). \( n = 6 \).

Step2: Calculate sums

• \( \sum x = 78.8 + 80.2 + 56.3 + 67.2 + 76.7 + 38.0 = 397.2 \)
• \( \sum y = 5.4 + 8.6 + 3.2 + 1.8 + 10.9 + 0.1 = 30.0 \)
• \( \sum xy = (78.8×5.4)+(80.2×8.6)+(56.3×3.2)+(67.2×1.8)+(76.7×10.9)+(38.0×0.1) \)

\( = 425.52 + 689.72 + 180.16 + 120.96 + 836.03 + 3.8 = 2256.19 \)

• \( \sum x^2 = 78.8^2 + 80.2^2 + 56.3^2 + 67.2^2 + 76.7^2 + 38.0^2 \)

\( = 6209.44 + 6432.04 + 3169.69 + 4515.84 + 5882.89 + 1444.0 = 27653.9 \)

• \( \sum y^2 = 5.4^2 + 8.6^2 + 3.2^2 + 1.8^2 + 10.9^2 + 0.1^2 \)

\( = 29.16 + 73.96 + 10.24 + 3.24 + 118.81 + 0.01 = 235.42 \)

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: \( 6×2256.19 - 397.2×30 = 13537.14 - 11916 = 1621.14 \)
• Denominator part 1: \( 6×27653.9 - 397.2^2 = 165923.4 - 157767.84 = 8155.56 \)
• Denominator part 2: \( 6×235.42 - 30^2 = 1412.52 - 900 = 512.52 \)
• Denominator: \( \sqrt{8155.56×512.52} \approx \sqrt{4180000} \approx 2044.5 \) (approximate, precise: \( \sqrt{8155.56×512.52} = \sqrt{4180000.0} \) (exact calculation: \( 8155.56×512.52 = 4180000.0 \) (approx))
• \( r = \frac{1621.14}{2044.5} \approx 0.793 \) (rounded to three decimals)

Answer:

\( 0.793 \)