- Complete a chi-square test for goodness of fit and write its conclusion in context of the problem
Conditions
The chi-square model is a family of curves that depend on degrees of freedom (number of categories minus 1). All chi-square curves are skewed to the right with a mean equal to the degrees of freedom.
- Random: Observed counts must come from a random sample (to ensure our conclusions are free from sampling bias).
- 10%: The sample size must be less than a tenth of the population size (to satisfy independence assumptions).
- Large Sample: The sample is large enough such that the expected counts are all five or greater (to ensure our sampling distribution resembles a chi-square distribution).
[latex]\chi^2[/latex] Test Statistic, P-Value, & Conclusions
If these conditions are met, we use the chi-square distribution to find the P-value. We use the same logic that we use in all hypothesis tests to draw a conclusion based on the P-value. If the P-value is at least as small as the significance level, we reject the null hypothesis and accept the alternative hypothesis.
The P-value is the likelihood that results from random samples have a [latex]\chi^2[/latex] value equal to or greater than that calculated from the data if the null hypothesis is true. For different degrees of freedom, the same [latex]\chi^2[/latex] value gives different P-values.
Using our mock data set, let’s draw a conclusion and make an inference regarding the State of New York claims that each county receives a number of vaccines that is proportional to its population size.
Step 2: Under “Enter Data,” choose “Contingency Table.”
Step 3: Change each of the counts to the observed counts.
Step 4: Change each of the “props” to the expected proportions.
| County | Queens | Bronx | Westchester |
| Observed Count | 204 | 132 | 164 |
| % | 48.8% | 30.6% | 20.6% |
Step 5: Then press “Submit.”