To see how a chi-square hypothesis test works with a multinomial experiment, we will investigate the following two examples.

### Example 1: A Fair Coin

A fair coin has a equal probability of 1/2 of coming up heads or tails. We toss a coin 1000 times and record the results of a total of 580 heads and 420 tails. We want to test the hypothesis at a 95% level of confidence that the coin we flipped is fair. More formally, the null hypothesis *H*_{0} is that the coin is fair. Since we are comparing observed frequencies of results from a coin toss to the expected frequencies from an idealized fair coin, a chi-square test should be used.

### Compute the Chi-Square Statistic

We begin by computing the chi-square statistic for this scenario. There are two events, heads and tails. Heads has an observed frequency of *f*_{1} = 580 with expected frequency of *e*_{1} = 50% x 1000 = 500. Tails has an observed frequency of *f*_{2} = 420 with expected frequency of *e*_{1} = 500.

We now use the formula for the chi-square statistic and see that χ^{2} = (*f*_{1} - *e*_{1} )^{2}/*e*_{1} + (*f*_{2} - *e*_{2} )^{2}/*e*_{2}= 80^{2}/500 + (-80)^{2}/500 = 25.6.

### Find the Critical Value

Next we need to find the critical value for the proper chi-square distribution. Since there are two outcomes for the coin there are two categories to consider. The number of degrees of freedom is one less than the number of categories: 2 - 1 = 1. We use the chi-square distribution for this number of degrees of freedom and see that χ^{2}_{0.95}=3.841.

### Reject or Fail to Reject?

Finally we compare the calculated chi-square statistic with the critical value from the table. Since 25.6 > 3.841, we reject the null hypothesis that this is a fair coin.

### Example 2: A Fair Die

A fair die has a equal probability of 1/6 of rolling a one, two, three, four, five or six. We roll a die 600 times and note that we roll a one 106 times, a two 90 times, a three 98 times, a four 102 times, a five 100 times and a six 104 times. We want to test the hypothesis at a 95% level of confidence that we have a fair die.

### Compute the Chi-Square Statistic

There are six events, each with expected frequency of 1/6 x 600 = 100. The observed frequencies are *f*_{1} = 106, *f*_{2} = 90, *f*_{3} = 98, *f*_{4} = 102, *f*_{5} = 100, *f*_{6} = 104,

We now use the formula for the chi-square statistic and see that χ^{2} = (*f*_{1} - *e*_{1} )^{2}/*e*_{1} + (*f*_{2} - *e*_{2} )^{2}/*e*_{2}+ (*f*_{3} - *e*_{3} )^{2}/*e*_{3}+(*f*_{4} - *e*_{4} )^{2}/*e*_{4}+(*f*_{5} - *e*_{5} )^{2}/*e*_{5}+(*f*_{6} - *e*_{6} )^{2}/*e*_{6} = 1.6.

### Find the Critical Value

Next we need to find the critical value for the proper chi-square distribution. Since there are six categories of outcomes for the die, the number of degrees of freedom is one less than this: 6 - 1 = 5. We use the chi-square distribution for five degrees of freedom and see that χ^{2}_{0.95}=11.071.

### Reject or Fail to Reject?

Finally we compare the calculated chi-square statistic with the critical value from the table. Since the calculated chi-square statistic is 1.6