Chi Squared Tests: Goodness of fit test

Chi-Squared Goodness of Fit Test

• The Chi-Squared Goodness of Fit test is a statistical procedure that can be used to determine if the observed distribution of data matches an expected distribution.
• It’s a nonparametric method, which means it makes no assumptions about the properties of the population from which the sample was taken.

The Assumptions

• The observations must be independent. This means that the occurrence of outcomes does not affect the probability of future outcomes.
• The data should be in the form of frequencies or counts of occurrences.
• The expected frequencies for each category should be at least 5. This is sometimes referred to as the 5-count rule.

The Null and Alternative Hypotheses

• The null hypothesis (H0) for the goodness of fit test is that the observed frequencies come from the expected distribution.
• The alternative hypothesis (H1) is that the observed frequencies do not follow the expected distribution.

Computation of Chi-Squared Test Statistic

• The Chi-squared statistic is calculated by comparing the observed frequencies with the expected frequencies in each category.
• The formula to calculate the test statistic is: χ² = Σ [(O−E)² / E] where O represents the observed frequency and E is the expected frequency.

Degrees of Freedom

• The degrees of freedom for the goodness of fit test are calculated as the number of categories minus one. It’s denoted as df.
• The degrees of freedom are used as the parameter to the chi-squared distribution to interpret the test statistic.

Interpreting the Results

• The test statistic is compared with a critical value from the Chi-Squared Distribution Table with the appropriate degrees of freedom.
• If the test statistic is greater than the critical value, the null hypothesis is rejected, i.e., it can be concluded that the observed distribution significantly differs from the expected distribution.

Applications

• The Chi-Squared Goodness of Fit test has wide applications in scientific research to test mathematical models against real-world data.
• Care should be taken not to use the test inappropriately, as violations of the assumptions can lead to inaccurate conclusions.