The goodness-of-fit test helps determine if sample data follows a specific probability distribution. It uses a chi-square statistic to compare observed frequencies with expected frequencies based on the hypothesized distribution.

Interpreting the test results involves comparing the calculated test statistic to a critical value. If the statistic exceeds the critical value, we reject the null hypothesis, suggesting the data doesn't fit the specified distribution.

Goodness-of-Fit Test

Goodness-of-fit test for distributions

• Determines if sample data comes from a population with a specific probability distribution (normal, binomial, Poisson)
• Null hypothesis ($H_0$) states the data follows the specified distribution
• Alternative hypothesis ($H_a$) states the data does not follow the specified distribution
• Steps to perform the test:
  1. State the hypotheses
  2. Calculate expected frequencies for each category based on the hypothesized distribution
  3. Calculate the test statistic using observed and expected frequencies
  4. Determine degrees of freedom and critical value from the chi-square distribution
  5. Compare test statistic to critical value and decide to reject or fail to reject $H_0$
  6. Calculate the p-value to assess the strength of evidence against $H_0$

Test statistic calculation

• Chi-square formula calculates the test statistic:
  • $\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}$
    • $\chi^2$ represents the test statistic
    • $O_i$ represents observed frequency for category $i$
    • $E_i$ represents expected frequency for category $i$
    • $k$ represents the number of categories
• Expected frequencies calculated by multiplying total sample size by probability of each category according to hypothesized distribution
• Degrees of freedom for the test:
  • $df = k - 1 - m$
    • $k$ represents the number of categories
    • $m$ represents the number of parameters estimated from the data

Interpretation of chi-square results

• Goodness-of-fit test is a right-tailed test
  • $H_0$ rejected if test statistic greater than critical value
• Critical value determined using chi-square distribution table with calculated degrees of freedom and desired significance level (typically $\alpha = 0.05$)
• If test statistic greater than critical value:
  • Reject $H_0$
  • Sufficient evidence suggests data does not follow specified distribution
• If test statistic less than or equal to critical value:
  • Fail to reject $H_0$
  • Insufficient evidence to conclude data does not follow specified distribution

Additional Considerations

• Goodness-of-fit test is a nonparametric test, meaning it does not assume a specific underlying distribution for the data
• Used primarily for categorical data analysis
• Can be extended to analyze contingency tables for independence or homogeneity tests