Question

Exercise 1. Defense and wins. The data in the table below shows the wining percentage and the yards per game allowed in the 2014-15 season for a random sampling of teams. At α = 0.05, can you support the claim that there is a correlation between the winning percentage and the yards allowed?

Team	Winning percentage	Total yards
Baltimore Ravens	0.625	336.9
Cleveland Browns	0.438	366.1
Denver Broncos	0.750	305.2
Jacksonville Jaguars	0.188	370.8
New England Patriots	0.750	344.1
Oakland Raiders	0.188	357.6
Pittsburgh Steelers	0.688	353.4

Answer 1

The correlation coefficient between the wining percentage and total yard is obtained using the formula,

Team	Winning percentage, Y	Total yards, X	X^2	Y^2	XY
Baltimore Ravens	0.625	336.9	113501.6	0.390625	210.5625
Cleveland Browns	0.438	366.1	134029.2	0.191844	160.3518
Denver Broncos	0.75	305.2	93147.04	0.5625	228.9
Jacksonville Jaguars	0.188	370.8	137492.6	0.035344	69.7104
New England Patriots	0.75	344.1	118404.8	0.5625	258.075
Oakland Raiders	0.188	357.6	127877.8	0.035344	67.2288
Pittsburgh Steelers	0.688	353.4	124891.6	0.473344	243.1392
Sum	3.627	2434.1	849344.6	2.251501	1237.968

Now, the hypothesis test for significance of the correlation coefficient is performed in following steps,

Step 1: The null and alternative hypotheses are,

Step 2: The significance level,

The critical value for the t-statistic for degree of freedom = n-2=5

Step 3: The t-statistic is obtained using the formula,

Step 4:

Since the t-statistic is less than t-critical value, the null hypothesis is not rejected. Hence there is no significant correlation is between percentage of wining and yard per game allowed.

ΠΣΧΥΗΣΧΣΥ) Γ vΠΙΣ ΣΧΕΙn ΣΥ ΣΥ

X2-849344.6, X 2434.1 Y 3.627 Y2 2.251501 XY 12237.968

ΙΣΧΥ ΣΧ ΣΥ) 162.7068 -0.70284 ν.ΣΧ ΣΧΡΙΙΣΥΣΥΡΙ 231.49856

Ho population correlation coefficient, p 0

HA population correlation coefficient, p0

0.05

teritical = 2.571 for 2-tailed

rVn 2 t V1-2

-0.70284 7- 2 -2.20933 1-(-0.70284)

t-statistic2.20933<teritical 2.571 _