Question

members of the popular band drumsolo are concerned that the songs on their most recent album are too long. they decide to see if there was any relationship between song length and popularity of their previous albums. for every previously released album, they measure the average song length (in minutes), x, and also note the number of albums sold, y.
average song length albums sold
3.90 186,271
4.69 146,483
4.85 226,842
4.96 235,511
6.71 66,461
find the correlation coefficient, r, of the data described above. record your answer to the nearest thousandth.

Explanation:

Step1: Recall correlation - coefficient formula

The correlation - coefficient \(r\) formula for a set of \(n\) data points \((x_i,y_i)\) is \(r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\sqrt{n\sum_{i = 1}^{n}y_i^{2}-(\sum_{i = 1}^{n}y_i)^{2}}}\). Let \(n = 5\), \(x=\{3.90,4.69,4.85,4.96,6.71\}\), \(y=\{186271,146483,226842,235511,66461\}\).

Step2: Calculate \(\sum_{i = 1}^{n}x_i\), \(\sum_{i = 1}^{n}y_i\), \(\sum_{i = 1}^{n}x_i^{2}\), \(\sum_{i = 1}^{n}y_i^{2}\) and \(\sum_{i = 1}^{n}x_iy_i\)

\(\sum_{i = 1}^{5}x_i=3.90 + 4.69+4.85+4.96+6.71 = 25.11\).
\(\sum_{i = 1}^{5}y_i=186271+146483+226842+235511+66461=861568\).
\(\sum_{i = 1}^{5}x_i^{2}=3.90^{2}+4.69^{2}+4.85^{2}+4.96^{2}+6.71^{2}=15.21 + 21.9961+23.5225+24.6016+45.0241 = 130.3543\).
\(\sum_{i = 1}^{5}y_i^{2}=186271^{2}+146483^{2}+226842^{2}+235511^{2}+66461^{2}\)
\(=34796785441+21457879289+51457992964+55464821121+4416064521=167591543336\).
\(\sum_{i = 1}^{5}x_iy_i=3.90\times186271+4.69\times146483+4.85\times226842+4.96\times235511+6.71\times66461\)
\(=726456.9+686005.27+1090183.7+1168134.56+445953.31=4126733.74\).

Step3: Substitute into the formula

\(n\sum_{i = 1}^{n}x_iy_i=5\times4126733.74 = 20633668.7\).
\(\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=25.11\times861568 = 21634972.48\).
\(n\sum_{i = 1}^{n}x_i^{2}=5\times130.3543 = 651.7715\).
\((\sum_{i = 1}^{n}x_i)^{2}=25.11^{2}=630.5121\).
\(n\sum_{i = 1}^{n}y_i^{2}=5\times167591543336=837957716680\).
\((\sum_{i = 1}^{n}y_i)^{2}=861568^{2}=742308098624\).

\(r=\frac{20633668.7 - 21634972.48}{\sqrt{651.7715 - 630.5121}\sqrt{837957716680 - 742308098624}}\)
\(=\frac{-1001303.78}{\sqrt{21.2594}\sqrt{95649618056}}\)
\(=\frac{-1001303.78}{\sqrt{21.2594\times95649618056}}\)
\(=\frac{-1001303.78}{\sqrt{2.03477\times10^{11}}}\)
\(=\frac{-1001303.78}{451084.2}\approx - 0.22\)

Answer:

\(-0.22\)