Although there were some changes in the roster from 2009 Red

Although there were some changes in the roster from 2009, Red Sox maintained many players, despite adding John Lackey for a sizable 18.7 million dollar contract. Were the average salaries of the Red Sox players remaining in the lineup affected as a result of this move? Use the data below the lists the 2009 and 2010 salaries (in millions of $) to construct an appropriate hypothesis test at the 0.05 significance level.

Player

2009 Salary

2010 Salary

Josh Beckett

11.1

12.1

J Drew

14.0

14.0

Jacoby Ellsbury

0.45

0.49

Jon Lester

1.0

3.7

Mike Lowell

14.5

14.5

Daisuke Matsuzaka

8.3

8.3

Hideki Okajimi

1.7

2.7

David Ortiz

13.0

13.0

Jonathan Papelbon

6.2

9.3

Dustin Pedroia

1.7

3.7

Jason Varitek

5.0

3.0

Tim Wakefield

4.0

3.5

Kevin Youkilis

6.2

9.3

Player	2009 Salary	2010 Salary
Josh Beckett	11.1	12.1
J Drew	14.0	14.0
Jacoby Ellsbury	0.45	0.49
Jon Lester	1.0	3.7
Mike Lowell	14.5	14.5
Daisuke Matsuzaka	8.3	8.3
Hideki Okajimi	1.7	2.7
David Ortiz	13.0	13.0
Jonathan Papelbon	6.2	9.3
Dustin Pedroia	1.7	3.7
Jason Varitek	5.0	3.0
Tim Wakefield	4.0	3.5
Kevin Youkilis	6.2	9.3

Solution

We use paired t test to do it because it tests the same subjects twice.

Let mu_d be the mean difference between 2009 Salary and 2010 Salary

The test hypothesis:

Null hypothesis: mu_d=0

Alternative hypothesis: mu_d not equal to 0

The test statistic is

t= mean difference/(s/vn)

=-0.80308/(1.53779/sqrt(13))

=-1.88

The degree of freedom =n-1=13-1=12

It is a two-tailed test.

So the p-value= 2*P(t with df=12 <-1.88) =0.0846 (from student t table)

Since the p-value is larger than 0.05, we do not reject the null hypothesis.

So we can not conclude that the average salaries of the Red Sox players were remaining in the lineup affected as a result of this move

6.70385	mean 2009 Salary
7.50692	mean 2010 Salary
-0.80308	mean difference (2009 Salary - 2010 Salary)
1.53779	std. dev.
0.42651	std. error
13	n
12	df