Testing the goodness of fit is a statistical procedure used to determine how well an observed set of data fits a theoretical or expected distribution.

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This type of test assesses whether there are significant differences between the observed data and what would be expected under a certain theoretical model or hypothesis.

**Chi-Square Test for Goodness of Fit:**

One commonly used method for testing goodness of fit is the chi-square test. The chi-square ((\chi^2)) statistic is calculated by comparing the observed frequencies in each category or interval with the expected frequencies under the theoretical distribution.

The formula for the chi-square statistic is:

[ \chi^2 = \sum \frac{{(O_i – E_i)^2}}{{E_i}} ]

Where:

- (O_i) is the observed frequency in category (i).
- (E_i) is the expected frequency in category (i).
- The sum is taken over all categories.

The chi-square statistic follows a chi-square distribution with degrees of freedom equal to the number of categories minus 1.

**Steps for Chi-Square Goodness of Fit Test:**

**Formulate Hypotheses:**

- Null Hypothesis ((H_0)): The observed data follows the expected distribution.
- Alternative Hypothesis ((H_a)): The observed data differs significantly from the expected distribution.

**Determine Expected Frequencies:**

- Calculate the expected frequencies under the assumed distribution.

**Calculate Chi-Square Statistic:**

- Use the formula to compute the chi-square statistic.

**Compare with Critical Value or P-Value:**

- Determine the critical value from the chi-square distribution table or calculate the p-value associated with the chi-square statistic.

**Make a Decision:**

- If the p-value is less than the significance level ((\alpha)), reject the null hypothesis. If the chi-square statistic is greater than the critical value, reject the null hypothesis.

**Example:**

Suppose you have observed frequencies of different eye colors in a sample and want to test whether these frequencies match the expected distribution in the population.

**Formulate Hypotheses:**

- (H_0): The observed distribution matches the expected distribution.
- (H_a): The observed distribution differs from the expected distribution.

**Determine Expected Frequencies:**

- Use population percentages for each eye color to calculate expected frequencies.

**Calculate Chi-Square Statistic:**

- Use the formula to calculate the chi-square statistic.

**Compare with Critical Value or P-Value:**

- Compare the chi-square statistic to the critical value or calculate the p-value.

**Make a Decision:**

- If the p-value is small (typically less than 0.05), or if the chi-square statistic is greater than the critical value, reject the null hypothesis.

The chi-square test for goodness of fit is widely used in various fields, including biology, psychology, and quality control, to assess whether observed data conforms to expected distributions.