Question

find the correlation coefficient, r, of the data described below.

laura is a pr agent for an up - and - coming band. she wants to convince the band members that being active on social media is good for their careers. she claims that a large social media following typically translates into ticket sales. to prove her point, lauren looked up information about several successful bands.

she compared the number of social media followers (in millions), x, to the average number of hours it takes these bands to sell out a concert, y.

followers (in millions) hours
1.26 160
1.64 168
5.41 143
7.90 159
8.24 121
round your answer to the nearest thousandth.
r =

Explanation:

Step1: List the data points

We have the following data points for \( x \) (followers in millions) and \( y \) (hours to sell out):
\( (x_1, y_1) = (1.26, 160) \)
\( (x_2, y_2) = (1.64, 108) \)
\( (x_3, y_3) = (5.41, 143) \)
\( (x_4, y_4) = (7.90, 159) \)
\( (x_5, y_5) = (8.24, 121) \)

Step2: Calculate the necessary sums

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

• \( \sum x = 1.26 + 1.64 + 5.41 + 7.90 + 8.24 = 24.45 \)
• \( \sum y = 160 + 108 + 143 + 159 + 121 = 691 \)
• \( \sum xy = (1.26 \times 160) + (1.64 \times 108) + (5.41 \times 143) + (7.90 \times 159) + (8.24 \times 121) \)

\( = 201.6 + 177.12 + 773.63 + 1256.1 + 997.04 = 3405.49 \)

• \( \sum x^2 = (1.26)^2 + (1.64)^2 + (5.41)^2 + (7.90)^2 + (8.24)^2 \)

\( = 1.5876 + 2.6896 + 29.2681 + 62.41 + 67.8976 = 163.8529 \)

• \( \sum y^2 = (160)^2 + (108)^2 + (143)^2 + (159)^2 + (121)^2 \)

\( = 25600 + 11664 + 20449 + 25281 + 14641 = 97635 \)

Step3: Use the correlation coefficient formula

The formula for the 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 \( n = 5 \) (number of data points).

First, calculate the numerator:
\( n\sum xy - \sum x \sum y = 5 \times 3405.49 - 24.45 \times 691 \)
\( = 17027.45 - 16894.95 = 132.5 \)

Next, calculate the denominator:
First, calculate \( n\sum x^2 - (\sum x)^2 = 5 \times 163.8529 - (24.45)^2 \)
\( = 819.2645 - 597.8025 = 221.462 \)

Then, calculate \( n\sum y^2 - (\sum y)^2 = 5 \times 97635 - (691)^2 \)
\( = 488175 - 477481 = 10694 \)

Now, take the square root of the product of these two:
\( \sqrt{221.462 \times 10694} \approx \sqrt{2368300.628} \approx 1538.93 \)

Finally, calculate \( r \):
\( r = \frac{132.5}{1538.93} \approx 0.0861 \)

Answer:

\( r \approx 0.0861 \)