Question

the table shows the average number of hours of daylight per day for the last four months of the year. what is the correlation coefficient for the data in the table? monthly daylight hours month of the year average hours of daylight per day 9 12.37 10 11.37 11 10.57 12 10 -0.993 -0.791 0.791 0.993

Explanation:

Step1: Recall correlation - coefficient formula

The correlation coefficient \(r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\)
Let \(x\) be the month number (\(9, 10, 11, 12\)) and \(y\) be the average hours of daylight per day (\(12.37,11.37,10.57,10\))
\(n = 4\)
\(\sum x=9 + 10+11+12=42\)
\(\sum y=12.37+11.37+10.57+10 = 44.31\)
\(\sum xy=9\times12.37+10\times11.37+11\times10.57+12\times10=111.33+113.7+116.27+120 = 461.3\)
\(\sum x^{2}=9^{2}+10^{2}+11^{2}+12^{2}=81 + 100+121+144 = 446\)
\(\sum y^{2}=12.37^{2}+11.37^{2}+10.57^{2}+10^{2}=153.0169+129.2769+111.7249+100 = 494.0187\)

Step2: Calculate the denominator part 1

\(n\sum x^{2}-(\sum x)^{2}=4\times446-42^{2}=1784 - 1764=20\)

Step3: Calculate the denominator part 2

\(n\sum y^{2}-(\sum y)^{2}=4\times494.0187-44.31^{2}=1976.0748 - 1963.3761=12.6987\)

Step4: Calculate the denominator

\(\sqrt{20\times12.6987}=\sqrt{253.974}\approx15.9365\)

Step5: Calculate the numerator

\(n(\sum xy)-(\sum x)(\sum y)=4\times461.3-42\times44.31=1845.2-1861.02=-15.82\)

Step6: Calculate the correlation - coefficient

\(r=\frac{-15.82}{15.9365}\approx - 0.993\)

Answer:

\(-0.993\)