Question

In a shocking move, Roger ”The Rocket” Clemens decides to come out of retirement. Let x be the
number games he pitches in, and y be the number of hit batsmen per game.

x 1 3 6
y 0.4 1 1.5

(a) Construct a scatter plot.
(b) Compute the sample correlation coefficient r.
(c) How would you describe the linear correlation between the two variables?
(d) Find the equation of the least squares line and graph on your scatter plot.
(e) Use the least squares line to predict the number of hit batsmen if he pitches in 5 games.

Answer 1

a)

b)

The provided data are shown in the table below

X	Y
1	0.4
3	1
6	1.5

Also, the following calculations are needed to compute the correlation coefficient:

	X	Y	X*Y	X2	Y2
	1	0.4	0.4	1	0.16
	3	1	3	9	1
	6	1.5	9	36	2.25
Sum =	10	2.9	12.4	46	3.41

The correlation coefficient r is computed using the following expression:

where

In this case, based on the data provided, we get that

Therefore, based on this information, the sample correlation coefficient is computed as follows

which completes the calculation.

c) the correlation between the two is highly positive

d)

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

Therefore, we find that the regression equation is:

Y = 0.2474 + 0.2158 X

e) At X = 5

Y = 0.2474 + 0.2158 X

Y = 0.2474 + 0.2158*5

Y = 0.2474 + 1.079

Y = 1.3264