Question

Suppose you flip two coins 100 times. The result are 20 HH, 27 HT, 30 TH and 23 TT. Are the coins fair? Test at %5 significance level.

Answer 1

Solution:

Given: Two coins are flipped 100 times.

We have to test if coins are fair or unfair.

Level of significance = 5%

Step 1) State H0 and H1:

If coins are fair then each outcome has an equal chance.

That is: P(HH) = P( HT) = P(TH) = P(TT) = 1/4 = 0.25

Thus

H0: Two coins are fair , that is: P(HH) = P( HT) = P(TH) = P(TT) = 1/4 = 0.25

Vs

H1: Two coins are fair , that is: at least of the proportion is different from 0.25

Step 2) Test statistic:

Chi square test statistic for goodness of fit

Where

O_i = Observed Counts

E_i =Expected Counts

Thus we need to make following table

Outcomes	Oi: Observed frequency	Expected proportions	Ei: Expected Frequency	Oi2/Ei
HH	20	0.25	25	16
HT	27	0.25	25	29.16
TH	30	0.25	25	36
TT	23	0.25	25	21.16
	N = 100

Thus

Step 3) Chi-square critical value:

df = k - 1 = 4 - 1 = 3

Level of significance = 0.05

Chi-square critical value = 7.815

Step 4) Decision Rule:

Reject null hypothesis H0, if Chi square test statistic > Chi-square critical value = 7.815, otherwise we fail to reject H0.
Since Chi square test statistic = < Chi-square critical value = 7.815, we fail to reject H0.

Step 5) Conclusion:

At 0.05 level of significance, we conclude that: Two coins are fair.