The following table lists World Series Lengths for the fifty years from 1926 to

Question

The following table lists World Series Lengths for the fifty years from 1926 to 1975. Test at the 0.10 level whether these data are compatible with the model that each World Series game is an independent Bernoulli trial with p = P(AL wins ) = P (NL wins ) = 1/2 .

Attempt:

I just need help on finding the probabilities.

For example , let X denote the lenght of a World Series.

1) P(X=4)P(X=4)= P (AL and NL wins 4 times ) =2(1/2)2

2) P(X=5)=(n5)(1/2)5(11/2)n5 I don't even know if this is correct.

3) P(X=6)

I need help on finding probabilities when X=5 and X=6. I think I need to use the binomial distribution. I don't know what to do. Can someone please help me? and show the formula and help me understand.

For, P(X=7)=1P(X4)P(X5)P(X6). I need part 2) and 3).

Thank you in advance

Explanation / Answer

let X denote the lenght of a World Series.

Here X ~ Binomial(n=50,p = 1/2)

The pmf of X=x is,

P(X=x) = (50 C x) * 1/2^x * (1 - 1/2)^(50-x) x = 0,1,2,3.........50

We can find probabilities upto x=50.

For ex. to find P(X=5)

P(X=5) = (50 C 5) * (1/2)^5 * (1/2)^(50-5) = 1.88E-09 = 0.0000

SImilarly to find P(X=6).

P(X=6) = (50 C 6) * (1/2)^6 * (1/2)^(50-6) = 1.41E^-08 = 0.0000

Similarly we can find probabilities for all the x values 0,1,2,3,............50

Note that the sum of the probabilities is equal to 1.

And if you have to test the hypothesis that,

H0 : Data follows binomial or Bernoulli distribution

H1 : Data does not follows binomial or Bernoulli distribution

Then we need observed and expected frequencies.

Expected frequncies we can find using formula:

E = n*p(x)

where n = 50

p(x) is the probability.

Also note that the sum of probability is equal to sum of frequency.

Then find the test statistic as,

X2 = (O - E)2 / E

where O is observed frequency and

E is expected frequency

Now find P-value or critical value for taking decision

P-value we can find by using EXCEL.

syntax :

=CHIDIST(x,deg_freedom)

where x is test statistic

deg_freedom = k-1

where k are number of categories

assume alpha = 0.10

If P-value < alpha then reject H0 at 0.10 level of significance

Otherwise accept H0.

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