Describe a situation where a GoodnessofFit Test could be use

Describe a situation where a Goodness-of-Fit Test could be used. Include as much detail as possible with this example. Explain why this test is appropriate.

Solution

Consider a standard package of milk chocolate M&Ms.

There are six different colors: red, orange, yellow, green, blue and brown. Suppose that we are curious about the distribution of these colors and ask, do all six colors occur in equal proportion? This is the type of question that can be answered with a goodness of fit test.

Setting

We begin by noting the setting and why the goodness of fit test is appropriate. Our variable of color is categorical. There are six levels of this variable, corresponding to the six colors that are possible. We will assume that the M&Ms we count will be a simple random sample from the population of all M&Ms.

Null and Alternative Hypotheses

The null and alternative hypotheses for our goodness of fit test reflect the assumption that we are making about the population.

Since we are testing whether the colors occur in equal proportions, our null hypothesis will be that all colors occur in the same proportion. More formally, if p1 is the population proportion of red candies, p2 is the population proportion of orange candies, and so on, then the null hypothesis is that p1 = p2 = . . . = p6 = 1/6.

The alternative hypothesis is that at least one of the population proportions is not equal to 1/6.

Actual and Expected Counts

The actual counts are the number of candies for each of the six colors. The expected count refers to what we would expect if the null hypothesis were true. We will let n be the size of our sample. The expected number of red candies is p1 n or n/6. In fact, for this example, the expected number of candies for each of the six colors is simply n times pi, or n/6.