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Carrying Out a Chi-Square Test for Goodness of Fit

The Chi-Square Test for Goodness of Fit is a powerful statistical tool used to determine if an observed distribution of a categorical variable differs significantly from a hypothesized or expected distribution. Before carrying out the test, it’s essential to have properly Setting Up a Chi-Square Goodness of Fit Test by stating hypotheses and defining parameters.

Conditions for Inference

Before calculating the test statistic and p-value, we must verify that the conditions for performing a Chi-Square Goodness of Fit test are met. These are:

1. Random: The data must come from a random sample or a randomized experiment. This ensures that the sample is representative of the population.
2. Independent:
   • The individual observations within the sample must be independent of each other.
   • When sampling without replacement, the sample size $ n $ must be less than $ 10% $ of the population size ( $ N $ ). That is, $ n \le 0.10N $ .
3. Large Counts (Expected Counts): All expected counts must be at least 5. This condition ensures that the $ \chi^2 $ distribution is a good approximation for the sampling distribution of the test statistic. If this condition is not met for any category, categories may need to be combined, which can alter the hypotheses.

Calculating the Chi-Square Test Statistic

The Chi-Square ( $ \chi^2 $ ) test statistic measures how far the observed counts deviate from the expected counts. The formula is:

$$ \chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}} $$
Where:

• Observed (O): The actual count for each category from the sample.
• Expected (E): The count that would be expected for each category if the null hypothesis were true. These are calculated as $ E = n \times p_i $ , where $ n $ is the total sample size and $ p_i $ is the hypothesized proportion for that category. Expected Counts in Two-Way Tables is a related concept for two-way tables, but here we use the hypothesized proportions.
• ** $ \sum $ **: Sum of these values over all categories.

A larger $ \chi^2 $ value indicates a greater discrepancy between the observed and expected distributions, providing stronger evidence against the null hypothesis.

Degrees of Freedom

The degrees of freedom ( $ df $ ) for a Chi-Square Goodness of Fit test are calculated as:

$$ df = \text{number of categories} - 1 $$
The degrees of freedom determine the specific shape of the $ \chi^2 $ distribution used to calculate the p-value.

Obtaining the P-Value

Once the $ \chi^2 $ test statistic and degrees of freedom are calculated, the p-value can be found. The p-value is the probability of observing a $ \chi^2 $ test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

Since the Chi-Square test is always right-tailed (larger $ \chi^2 $ values indicate stronger evidence against $ H_0 $ ), the p-value is the area under the $ \chi^2 $ distribution curve to the right of the calculated test statistic.

• You can use a calculator’s $ \chi^2 $ cdf function (e.g., $ \chi^2 $ cdf(lower_bound = $ \chi^2 $ _statistic, upper_bound = E99, df)).
• Alternatively, statistical software will provide the p-value directly.

A small p-value (typically less than the significance level $ \alpha $ ) suggests that the observed differences are unlikely to have occurred by chance if the null hypothesis were true, leading to rejection of $ H_0 $ . Interpreting p-Values provides more detail on this.

Conclusion

The final step is to make a conclusion in the context of the problem, linking back to the null and alternative hypotheses. This involves comparing the p-value to the chosen significance level $ \alpha $ .

• **If p-value $ < \alpha $ **: We reject the null hypothesis ( $ H_0 $ ). There is convincing evidence that the observed distribution of the categorical variable is significantly different from the hypothesized distribution.
• **If p-value $ \ge \alpha $ **: We fail to reject the null hypothesis ( $ H_0 $ ). There is not convincing evidence that the observed distribution of the categorical variable is significantly different from the hypothesized distribution.

Remember that failing to reject $ H_0 $ does not mean we accept $ H_0 $ ; it simply means we don’t have enough evidence to claim a difference. Concluding a Test for a Population Proportion offers a general framework for drawing conclusions in hypothesis tests.