Question

Consider the following bivariate data. Point A B C D E F G H I J x 0 6 5 6 0 0 2 5 1 0 y 5 3 1 5 3 6 4 6 5 0 (a) Construct a scatter diagram of the given bivariate data. (Do this on paper. Your instructor may ask you to turn in this work.) (b) Calculate the covariance. (Give your answer correct to two decimal places.) (c) Calculate sx and sy. (Give your answers correct to three decimal places.) sx = sy =

(d) Calculate r using formula 13.2. (Give your answer correct to two decimal places.) r= covar(x,y)/(sx times sy)

(e) Calculate r using formula 13.3. (Give your answer correct to two decimal places.) r=SS(xy)/ sqrt of SS(x) times SS(y)

Answer 1

The given Bivariate data is

x <- c(0, 6, 5 ,6 ,0 ,0 ,2, 5 ,1 ,0)
y <- c(5 ,3 ,1, 5 ,3 ,6, 4, 6, 5, 0)

a)The scatter plot is given below.

Scatterplot 0 4 0 6

b) The covariance is calculated as

$Cov\left ( x,y \right )=\frac{\sum_{i=1}^{10}\left ( x_i-\overline{x} \right )\left ( y_i-\overline{y} \right )}{10-1}\\ {\color{Blue} Cov\left ( x,y \right )=0.1111111}$

c) The sample variances are calculated as

$s^2_x=\frac{\sum_{i=1}^{10}\left ( x_i-\overline{x} \right )^2}{10-1}\\ {\color{Blue} s_x=2.677}\\ s^2_y=\frac{\sum_{i=1}^{10}\left ( y_i-\overline{y} \right )^2}{10-1}\\ {\color{Blue} s_y=2.044}$

d) The correlation coefficient is

$r=\frac{Cov(x,y)}{s_xs_y}\\ r=\frac{0.1111}{2.677\times 2.044}\\ {\color{Blue} r=0.02}$

e)

$r=\frac{\sum_{i=1}^{10}\left ( x_i-\overline{x} \right )\left ( y_i-\overline{y} \right )}{\sqrt{\sum_{i=1}^{10}\left ( x_i-\overline{x} \right )^2\sum_{i=1}^{10}\left ( y_i-\overline{y} \right )^2}}\\ {\color{Blue} r=0.02}$