Question

Question 1

Part (a) A few years ago, Mars sold M&Ms inspired by candy corn. The M&Ms in those bags came in three colors: white, yellow, and orange. The following table shows the distributions of those colors in a bag of 90 M&Ms purchased at a local store.

Color of M&Ms	Count
White	32
Yellow	33
Orange	25

Using an alpha of 0.05, carry out a goodness-of-fit test for the null hypothesis that the three colors appear equally often.

(Hint: We will not be able to reject the null hypothesis, but you need to show the formal steps and calculations that lead to that decision.)

Part (b) State one example for the distribution of the three colors of M&Ms that would lead us to reject the null hypothesis that the three colors appear equally often. You again need to show the formal steps and calculations that lead to that decision.

Answer 1

a) The expected frequency along with the original data is given below:

Color of M&Ms	Count	Expected
White	32	30
Yellow	33	30
Orange	25	30

Hence the value of the test statistic is

Now observe that the fitted distribution is discrete uniform distribution, which says that the number of parameters needed is 1. Hence the degree of chi square is 1. Hence the critical value of the test statistic with 5% level of confidence is 3.84. Since the obtained value of test statistic is lesser than the critical value, hence we can not reject the null hypothesis at 5% confidence level.

b) Now consider the following frequency distribution with the expected frequency:

Color of M&Ms	Count	Expected
White	90	30
Yellow	0	30
Orange	0	30

In this case the value of the test statistic is = (90-30)²/30 = 30, which is greater than the critical value of the test statistic at 5% confidence level. Hence in this frequency distribution, we can reject the null hypothesis.