• Complete a chi-square test for goodness of fit and write its conclusion in context of the problem

[latex]\chi^2[/latex] test for goodness of fit

A goodness-of-fit hypothesis test determines whether or not the distribution of a categorical variable in a sample fits a claimed distribution in the population.

We can answer the following research questions with a chi-square goodness-of-fit test:

• According to the manufacturer of M&M candy, the color distribution for plain chocolate M&Ms is [latex]13\%[/latex] brown, [latex]13\%[/latex] red, [latex]14\%[/latex] yellow, [latex]24\%[/latex] blue, [latex]20\%[/latex] orange, and [latex]16\%[/latex] green. Do the M&Ms in our sample suggest that the color distribution is different?

• The distribution of blood types for whites in the United States is [latex]45\%[/latex] type O, [latex]41\%[/latex] type A, [latex]10\%[/latex] type B, and [latex]4\%[/latex] type AB. Is the distribution of blood types different for Asian Americans?

The null hypothesis states a specific distribution of proportions for each category of the variable in the population.
The alternative hypothesis says that the distribution is different from that stated in the null hypothesis.
To test our hypotheses, we select a random sample from the population and determine the distribution of the categorical variable in the data.

Let’s revisit the Italian football scenario. Recall that researchers measured birth rates in Italy and found the following results:

Quarter	Quarter 1 (Jan. – March)	Quarter 2 (April – June)	Quarter 3 (July – Sept.)	Quarter 4 (Oct. – Dec.)
Proportion of births in Italy	[latex]22.48\%[/latex]	[latex]24.98\%[/latex]	[latex]25.74\%[/latex]	[latex]26.80\%[/latex]

STEPS for Hypothesis Testing:

1. Write out the null and alternative hypotheses.
2. Check the conditions/assumptions.
3. Calculate a test statistic.
4. Calculate a P-value.
5. Compare the P-value to the significance level, [latex]\alpha[/latex], to make a decision.
   

   Decision	Conclusion
   If P-value [latex]\le\alpha[/latex], there is enough evidence to reject the null hypothesis.	At the [latex]\alpha\times[/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.
   If P-value [latex]\gt\alpha[/latex], there is not enough evidence to reject the null hypothesis.	At the [latex]\alpha\times[/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.

6. Write a conclusion in context (e.g., we do/do not have convincing evidence…).