Question

Consider the following set of data.

(19, 11), (26, 53), (65, 28), (77, 20), (108, 56), (118, 11)

(a) Calculate the covariance of the set of data. (Give your answer correct to two decimal places.)
A = -5.63

(b) Calculate the standard deviation of the six x-values and the standard deviation of the six y-values. (Give your answers correct to three decimal places.)

sx

=

sy

=

(c) Calculate r, the coefficient of linear correlation, for the data in part (a). (Give your answer correct to two decimal places.)

Answer 1

ANSWER:-

$X$	$Y$	$X-\bar{X}$	$Y-\bar{Y}$	$(X-\bar{X})(Y-\bar{Y})$
$19$	$11$	$(19-68.83)=-49.83$	$(11-29.83)=-18.83$	$(-49.83*-18.83)=938.2989$
$26$	$53$	$(26-68.83)=-42.83$	$(53-29.83)=23.17$	$(-42.83*23.17)=-992.3711$
$65$	$28$	$(65-68.83)=-3.83$	$(28-29.83)=-1.83$	$(-3..83*-1.83)=7.0089$
$77$	$20$	$(77-68.83)=8.17$	$(20-29.83)=-9.83$	$(8.17*-9.83)=-80.3111$
$108$	$56$	$(108-68.83)=39.17$	$(56-29.83)=26.17$	$(39.17*26.17)=1025.0789$
$118$	$11$	$(118-68.83)=49.17$	$(11-29.83)=-18.83$	$(49.17*-18.83)=-925.8711$

$\bar{X}=\frac{19+26+65+77+108+118}{6}$

$\bar{X}=\frac{413}{6}=68.8333$

$\bar{X}=68.83$

$\bar{Y}=\frac{11+53+28+20+56+11}{6}$

$\bar{Y}=\frac{179}{6}$

$\bar{Y}=29.83$
$calculation of Cov(X,Y):-$

$We know that$

$Cov(X,Y)=\frac{\sum (X-\bar{X})(Y-\bar{Y})}{n-1}$

$Cov(X,Y)=\frac{938.2989-992.3711+7.0089-80.3111+1025.0789-925.8711}{5}$

$Cov(X,Y)=\frac{-28.1666}{5}$

$Cov(X,Y)=-5.633$

$Calculation of Sx:-$

$Sx=\sqrt{\frac{\sum (X-\bar{X})^2}{n}}$

$Sx=\sqrt{\frac{(-49.83)^2+(-42.83)^2+(-3.83)^2+(8.17)^2+(39.17)^2+(49.17)^2)}{6}}$

$Sx=\sqrt{\frac{2483.0289+1834.4089+14.6689+66.7489+1534.2889+2417.6889}{6}}$

$Sx=\sqrt{\frac{8350.8334}{6}}$

$Sx=\sqrt{1391.805567}$

$Sx=37.3069$

$Sx=37.307$

$Calculation of Sy:-$

$Sy=\sqrt{\frac{\sum (Y-\bar{Y})^2}{n}}$

$Sy=\sqrt{\frac{(18.83)^2+(23.17)^2+(-1.83)^2+(-9.83)^2+(26.17)^2+(-18.83)^2)}{6}}$

$Sy=\sqrt{\frac{354.5689+536.8489+3.3489+96.6289+684.8689+354.5689}{6}}$

$Sy=\sqrt{\frac{2030.8334}{6}}$

$Sy=\sqrt{338.4722}$

$Sy=18.3976$

$Sy=18.398$

$calculation of r:-$

$We know that$

$r=\frac{Cov(X,Y)}{SxSy}$

$Where,$

$Cov(X,Y)=-5.633$

$Sx=37.307$

$Sy=18.398$

$r=-8.2074*10^-3$

$r=-0.00821$