Question

question 14, 9.1.37
calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values. then calculate it again, letting row 2 represent the x - values and row 1 the y - values. what effect does switching the variables have on r?
row 1: 19 20 37 46 60 63
row 2: 132 172 152 169 113 109
calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values.
r = (round to three decimal places as needed.)

Explanation:

Step1: Calculate sums

Let $x$ be values from Row 1: $x_1 = 19,x_2 = 20,x_3 = 37,x_4 = 46,x_5 = 60,x_6 = 63$, and $y$ be values from Row 2: $y_1 = 132,y_2 = 172,y_3 = 152,y_4 = 169,y_5 = 113,y_6 = 109$.
$n = 6$.
$\sum_{i = 1}^{n}x_i=19 + 20+37 + 46+60+63=245$
$\sum_{i = 1}^{n}y_i=132 + 172+152+169+113+109 = 847$
$\sum_{i = 1}^{n}x_i^2=19^2+20^2+37^2+46^2+60^2+63^2=361 + 400+1369+2116+3600+3969 = 11815$
$\sum_{i = 1}^{n}y_i^2=132^2+172^2+152^2+169^2+113^2+109^2=17424+29584+23104+28561+12769+11881 = 123323$
$\sum_{i = 1}^{n}x_iy_i=19\times132+20\times172+37\times152+46\times169+60\times113+63\times109=2508+3440+5624+7774+6780+6867 = 33993$

Step2: Calculate numerator of $r$

$S_{xy}=\sum_{i = 1}^{n}x_iy_i-\frac{\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n}=33993-\frac{245\times847}{6}=33993-\frac{207515}{6}\approx33993 - 34585.83=- 592.83$

Step3: Calculate denominator of $r$

$S_{xx}=\sum_{i = 1}^{n}x_i^2-\frac{(\sum_{i = 1}^{n}x_i)^2}{n}=11815-\frac{245^2}{6}=11815-\frac{60025}{6}\approx11815 - 10004.17 = 1810.83$
$S_{yy}=\sum_{i = 1}^{n}y_i^2-\frac{(\sum_{i = 1}^{n}y_i)^2}{n}=123323-\frac{847^2}{6}=123323-\frac{717409}{6}\approx123323 - 119568.17 = 3754.83$
$S_{xx}S_{yy}=1810.83\times3754.83\approx6799997.7$
$\sqrt{S_{xx}S_{yy}}\approx2607.68$

Step4: Calculate $r$

$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{- 592.83}{2607.68}\approx - 0.227$

If we switch $x$ and $y$, the formula for the correlation - coefficient $r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i - \bar{y})^2}}$ is symmetric in $x$ and $y$. So the value of $r$ remains the same.

Answer:

$-0.227$