Question

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

Answer 1

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.