In: Statistics and Probability
For a sports radio talk show, you are asked to research the question whether more home runs are hit by players in the National League or by players in the American League. You decide to use the home run leaders from each league for the last 40 years as your data.
National League
47 49 73 50 65 70 49 47 40 43
46 35 38 40 47 39 49 37 37 36
40 37 31 48 48 45 52 38 38 36
44 40 48 45 45 36 39 44 52 47
American League
47 57 52 47 48 56 56 52 50 40
46 43 44 51 36 42 49 49 40 43
39 39 22 41 45 46 39 32 36 32
32 32 37 33 44 49 44 44 49 32
What are you hypotheses?
The significance level is α = 0.05
For a sports radio talk show, you are asked to research the question whether more home runs are hit by players in the National League or by players in the American League.
The given data for National League is
National League
47 49 73 50 65 70 49 47 40 43
46 35 38 40 47 39 49 37 37 36
40 37 31 48 48 45 52 38 38 36
44 40 48 45 45 36 39 44 52 47
The given data for American League is
American League
47 57 52 47 48 56 56 52 50 40
46 43 44 51 36 42 49 49 40 43
39 39 22 41 45 46 39 32 36 32
32 32 37 33 44 49 44 44 49 32
Here n1= n2 = n = 40 ( home run leaders from each league for the last 40 years )
What are you hypotheses?
Research the question whether more home runs are hit by players in the National League or by players in the American League
So hypothesi will be
H0 : 1 2 ( players in the National League hit more home runs )
H1 : 1 < 2 ( players in the National League do not hit more home runs than American League )
Now we will calculate sample mean and vaiance of both National League and American League
Sample mean =
Sample variance =
For National League
Sample mean = ( 47 +49+ 73 +50 +65+ 70+49 + ........+36 +39+44+ 52+ 47) /40
= 1790 / 40
= 44.75
Samplevariance s12
s12 =[ (47-44.75)2 +(49-44.75)2+ (73-44.75)2 +(50-44.75)2 +(65+ ........+(36-44.75)2 +(39-44.75)2+
(44-44.75)2+ 52-44.75)2+ 47-44.75)2] (40--1)
= 3077.5 / 39
s12 = 78.91026
Similarly we calculate sample mean and variance for American League
which are = 42.875 , s22 = 61.1891
Test Statistics T.S
T.S =
where S.E =
sp2 =
=
sp2 = 70.04968
Thus S.E =
S.E =1.871492
T.S =
=
T.S = 1.001874
Thus test statistics value is 1.001874
To calculate P-value
P-value = P( X < T.S )
= P( X < 1.001874 ) , where X ~ t(n1+n2-2= 78) i.e t78 distribution with 78 df
From R
>
pt(1.001874,df=78)
# P( X < 1.001874 )
[1] 0.8402488
Hence , P-value = P( X < 1.001874 )
i.e P-value = 0.8402488
Conclusion -
The P-value = 0.8402 is greater than 0.05 , hence at 5% of level do significance we do not reject null hypothesis .
So , there may be chance that more home runs are hited by players in the National League than players in the American League.