Describe the chi-square goodness-of-fit test.

http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm

If you cannot understand the above site, possibly this might help.

Use the Chi-square (X^2) method.

X^2 = ∑ (O-E)^2/E, where O = observed frequency and E = expected frequency.

∑ = sum of all the cells.

E = (column total * row total)/grand total

df = n - 1, where n = number of cells

Look up value in X^2 table in the back of your textbook.

The chi-square goodness-of-fit test is a statistical test used to determine whether an observed frequency distribution significantly differs from an expected frequency distribution. It is commonly used when comparing a categorical variable to a known distribution or a theoretical distribution.

To perform the chi-square goodness-of-fit test, follow these steps:

1. Hypothesis formulation: Formulate the null hypothesis (H0) and alternative hypothesis (Ha). The null hypothesis states that there is no significant difference between the observed and expected frequencies, while the alternative hypothesis suggests there is a significant difference.

2. Expected frequencies calculation: Calculate the expected frequencies for each category based on the sample size and the expected distribution. The expected distribution can be derived from a theoretical distribution or a known distribution.

3. Chi-square statistic computation: Calculate the chi-square test statistic using the formula:

chi-square = Σ((observed frequency - expected frequency)^2) / expected frequency

This formula compares the observed and expected frequencies for each category and calculates the sum of the squared differences divided by the expected frequency.

4. Degrees of freedom determination: Determine the degrees of freedom (df) for the test, which is equal to the number of categories minus 1. This reflects the number of independent pieces of information in the data.

5. Critical value determination: Look up the critical value from the chi-square distribution table at the desired level of significance and with the degrees of freedom.

6. Compare the test statistic and critical value: If the test statistic is greater than the critical value, reject the null hypothesis and conclude that there is a significant difference between the observed and expected frequencies. If the test statistic is smaller than the critical value, fail to reject the null hypothesis.

7. Interpretation of results: Based on the outcome of the test, interpret the results and draw conclusions about whether there is evidence for a significant difference between the observed and expected frequencies.

It is important to note that the chi-square goodness-of-fit test assumes that the observations are independent and the expected frequencies in each category are not too small.