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 | 78.3 78.5 55.9 67.9 77.2 37.8
award winners | 5.5 9.2 3.3 1.8 10.8 0.1
the linear correlation coefficient is r = \square
(round to three decimal places as needed.)

Explanation:

Step1: Identify Variables

Let \( x \) be Internet Users (per 100 people) and \( y \) be Award Winners (per 10 million people). The data points are:
\( (78.3, 5.5) \), \( (78.5, 9.2) \), \( (55.9, 3.3) \), \( (67.9, 1.8) \), \( (77.2, 10.8) \), \( (37.8, 0.1) \).

Step2: Calculate Summations

• \( \sum x = 78.3 + 78.5 + 55.9 + 67.9 + 77.2 + 37.8 = 395.6 \)
• \( \sum y = 5.5 + 9.2 + 3.3 + 1.8 + 10.8 + 0.1 = 30.7 \)
• \( \sum xy = (78.3 \times 5.5) + (78.5 \times 9.2) + (55.9 \times 3.3) + (67.9 \times 1.8) + (77.2 \times 10.8) + (37.8 \times 0.1) \)

\( = 430.65 + 722.2 + 184.47 + 122.22 + 833.76 + 3.78 = 2297.08 \)

• \( \sum x^2 = 78.3^2 + 78.5^2 + 55.9^2 + 67.9^2 + 77.2^2 + 37.8^2 \)

\( = 6130.89 + 6162.25 + 3124.81 + 4610.41 + 5959.84 + 1428.84 = 27417.04 \)

• \( \sum y^2 = 5.5^2 + 9.2^2 + 3.3^2 + 1.8^2 + 10.8^2 + 0.1^2 \)

\( = 30.25 + 84.64 + 10.89 + 3.24 + 116.64 + 0.01 = 245.67 \)

Step3: Apply Correlation Formula

The linear correlation coefficient \( r \) is given by:

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

where \( n = 6 \).

Substep3.1: Numerator

\( n\sum xy - (\sum x)(\sum y) = 6(2297.08) - (395.6)(30.7) \)
\( = 13782.48 - 12144.92 = 1637.56 \)

Substep3.2: Denominator Components

• \( n\sum x^2 - (\sum x)^2 = 6(27417.04) - (395.6)^2 \)

\( = 164502.24 - 156499.36 = 8002.88 \)

• \( n\sum y^2 - (\sum y)^2 = 6(245.67) - (30.7)^2 \)

\( = 1474.02 - 942.49 = 531.53 \)

Substep3.3: Denominator

\( \sqrt{8002.88 \times 531.53} \approx \sqrt{4253700} \approx 2062.45 \)

Step4: Calculate \( r \)

$$ r = \frac{1637.56}{2062.45} \approx 0.794 $$

Answer:

\( r \approx \boxed{0.794} \)