Question

In baseball, is there a linear correlation between batting average and home run percentage? Let x represent the batting average of a professional baseball player, and let y represent the player's home run percentage (number of home runs per 100 times at bat). A random sample of n = 7 professional baseball players gave the following information.

x 0.255 0.251 0.286 0.263 0.268 0.339 0.299

y 1.5 3.9 5.5 3.8 3.5 7.3 5.0

(a) Make a scatter diagram of the data. Then visualize the line you think best fits the data.

(b) Use a calculator to verify that Σx = 1.961, Σx2 = 0.555, Σy = 30.5, Σy2 = 152.69 and Σxy = 8.842. Compute r. (Round to 3 decimal places.)

As x increases, does the value of r imply that y should tend to increase or decrease? Explain your answer.

Given our value of r, we can not draw any conclusions for the behavior of y as x increases.

Given our value of r, y should tend to remain constant as x increases.

Given our value of r, y should tend to increase as x increases.

Given our value of r, y should tend to decrease as x increases.

Answer 1

Solution:

Part a

Required scatter diagram is given as below:

From above scatter diagram, it is observed that there is a positive linear relationship exists between the given two variables.

Part b

The formula for correlation coefficient is given as below:

Correlation coefficient = r = [n∑xy - ∑x∑y]/sqrt[(n∑x^2 – (∑x)^2)*(n∑y^2 – (∑y)^2)]

Required calculation table is given as below:

No.	x	y	x^2	y^2	xy
1	0.255	1.5	0.065025	2.25	0.3825
2	0.251	3.9	0.063001	15.21	0.9789
3	0.286	5.5	0.081796	30.25	1.573
4	0.263	3.8	0.069169	14.44	0.9994
5	0.268	3.5	0.071824	12.25	0.938
6	0.339	7.3	0.114921	53.29	2.4747
7	0.299	5	0.089401	25	1.495
Total	1.961	30.5	0.555137	152.69	8.8415

Σx = 1.961, Σx2 = 0.555, Σy = 30.5, Σy2 = 152.69 and Σxy = 8.842

r = [n∑xy - ∑x∑y]/sqrt[(n∑x^2 – (∑x)^2)*(n∑y^2 – (∑y)^2)]

r = [7*8.842- 1.961*30.5]/sqrt[(7*0.555 – (1.961)^2)*(7*152.69 – (30.5)^2)]

r = 0.878655

r = 0.879

There is a strong positive linear correlation or association exists between the given two variables.

Given our value of r, y should tend to increase as x increases.