13.1 Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test helps determine if categorical data matches a specific distribution. It compares observed frequencies to expected ones, calculating a test statistic to assess the fit between sample data and hypothesized population distribution.

Interpreting results involves comparing the test statistic to a critical value or using the p-value. This helps decide whether to reject or fail to reject the null hypothesis, indicating if the sample data significantly differs from the expected distribution.

Chi-Square Goodness-of-Fit Test

Purpose of chi-square goodness-of-fit test

• Determines if a sample of categorical data (colors, types) comes from a population with a specific distribution
• Compares observed frequencies of categories in the sample to expected frequencies based on the hypothesized distribution
• Applicable when data is categorical or nominal, sample is randomly selected, and expected frequency of each category is at least 5

Calculation of chi-square test statistic

• Calculate expected frequencies for each category by multiplying hypothesized probability of each category by total sample size
• Chi-square test statistic calculated using formula $\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}$
  • $O_i$ represents observed frequency for category $i$ (actual count)
  • $E_i$ represents expected frequency for category $i$ (calculated based on hypothesized distribution)
  • $k$ represents number of categories (options, choices)

Degrees of freedom for chi-square tests

• Degrees of freedom for chi-square goodness-of-fit test calculated as $df = k - 1$
  • $k$ represents number of categories
• Critical value determined by significance level ($\alpha$), degrees of freedom
  • Found using chi-square distribution table or statistical software (Excel, R)

Interpretation of chi-square test results

• Compare calculated chi-square test statistic to critical value
  1. If test statistic greater than critical value, reject null hypothesis
     • Suggests sample data does not follow hypothesized distribution (significantly different)
  2. If test statistic less than or equal to critical value, fail to reject null hypothesis
     • Suggests sample data consistent with hypothesized distribution (not significantly different)
• P-value also used to make decision
  • If p-value less than significance level ($\alpha$), reject null hypothesis
  • If p-value greater than or equal to significance level ($\alpha$), fail to reject null hypothesis