Question

what is the value of r for the data in the table?

	ukraine	ethiopia	guam	south africa	yemen	mexico	nigeria	macau
female life expectancy	74.77	59.21	81.73	48.45	66.27	79.63	55.33	87.54

data from central intelligence agency 2013
1
0.97
−0.97
0.94

Explanation:

Step1: Recall correlation - coefficient formula

The Pearson correlation coefficient $r$ formula 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 $n$ is the number of data - points, $x$ represents male life expectancy values, and $y$ represents female life expectancy values.

Step2: Calculate sums

Let $n = 8$. Calculate $\sum x$, $\sum y$, $\sum xy$, $\sum x^{2}$, and $\sum y^{2}$ from the data in the table.
Let $x_1 = 63.07,x_2 = 53.99,x_3 = 75.46,x_4 = 50.34,x_5 = 62.05,x_6 = 73.84,x_7 = 48.95,x_8 = 81.47$ and $y_1 = 74.77,y_2 = 59.21,y_3 = 81.73,y_4 = 48.45,y_5 = 66.27,y_6 = 79.63,y_7 = 55.33,y_8 = 87.54$.
$\sum x=63.07 + 53.99+75.46+50.34+62.05+73.84+48.95+81.47 = 509.17$
$\sum y=74.77 + 59.21+81.73+48.45+66.27+79.63+55.33+87.54 = 552.93$
$\sum xy=(63.07\times74.77)+(53.99\times59.21)+(75.46\times81.73)+(50.34\times48.45)+(62.05\times66.27)+(73.84\times79.63)+(48.95\times55.33)+(81.47\times87.54)$
$=4715.64+3207.75+6165.45+2437.97+4112.71+5889.98+2704.00+7130.74 = 36364.24$
$\sum x^{2}=63.07^{2}+53.99^{2}+75.46^{2}+50.34^{2}+62.05^{2}+73.84^{2}+48.95^{2}+81.47^{2}$
$=3978.82+2914.92+5694.21+2534.12+3849.20+5452.34+2396.10+6637.36 = 33456.07$
$\sum y^{2}=74.77^{2}+59.21^{2}+81.73^{2}+48.45^{2}+66.27^{2}+79.63^{2}+55.33^{2}+87.54^{2}$
$=5590.55+3506.82+6673.79+2347.40+4392.71+6340.94+3061.41+7664.35 = 39577.97$

Step3: Substitute into formula

$r=\frac{8\times36364.24-(509.17\times552.93)}{\sqrt{[8\times33456.07-(509.17)^{2}][8\times39577.97-(552.93)^{2}]}}$
$n(\sum xy)=8\times36364.24 = 290913.92$
$(\sum x)(\sum y)=509.17\times552.93=281563.07$
$n\sum x^{2}=8\times33456.07 = 267648.56$
$(\sum x)^{2}=509.17^{2}=259253.09$
$n\sum y^{2}=8\times39577.97 = 316623.76$
$(\sum y)^{2}=552.93^{2}=305721.58$
$r=\frac{290913.92 - 281563.07}{\sqrt{(267648.56 - 259253.09)(316623.76 - 305721.58)}}$
$=\frac{9350.85}{\sqrt{(8395.47)(10902.18)}}$
$=\frac{9350.85}{\sqrt{91527451.15}}$
$=\frac{9350.85}{9567.00}\approx0.97$

Answer:

0.97