Hypothesis testing using chi-square distribbution

Question

Four players meet weekly and play eight hands of cards. Over a year, one of the payers finds that he has won x of the eight hands with frequency fx given in the following table:

x 0 1 2 3 4 5 6 7 8

fx 4 13 12 12 6 3 2 0 0

Find the frequencies of the number of hands he would expect to win if the probability of winning any hand were 1/4, and if the outcomes of different hands were independent. Use the chi-square distribution to test this hypothesis.

My answer:

test statistics= 47.77

degrees of freedom= 13-1=12

testing at 1%, we get 26.217

47.77> 26.217, so reject Ho

I not sure if this is the correct answer, if I got it wrong, can anyone show me how to obtain the correct answer please?

Answer 1

I'm not going to re-calculate your work, but here's what will give you the right answer:

1. The probability of winning $x$ of 8 hands is distributed $binomial(x,8,0.25)$. Just calculate the expected number of times you expect to win a given number of hands by multiplying 8 by the probability calculated above.
2. Calculate your chi-square terms based on the expected number of wins above and the observed number of wins.
3. Correct for the number of bins used (here you used 9 bins..0 thru 8) so you have n-1-9 or n-10 df where n=52 so you have df=42