What is the primary use of the $\chi 2^{}$ goodness of fit test? Select one.

• A. To compare the means of two independent samples.
• B. To estimate the population variance from a sample.
• C. To test for a linear relationship between two continuous variables.
• D. To determine whether observed categorical data significantly differ from expected frequencies under a specific theoretical distribution.

Answer: (D)

Verified step by step guidance

1. Understand that the chi-square goodness of fit test is used to analyze categorical data, where data are divided into categories or groups.
2. Recognize that this test compares the observed frequencies (counts) in each category to the expected frequencies, which are derived from a theoretical distribution or hypothesis.
3. Formulate the null hypothesis \( H_0 \) stating that the observed frequencies fit the expected distribution, and the alternative hypothesis \( H_a \) stating that they do not fit.
4. Calculate the chi-square test statistic using the formula: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\] where \( O_i \) is the observed frequency and \( E_i \) is the expected frequency for category \( i \).
5. Compare the calculated \( \chi^2 \) value to the critical value from the chi-square distribution with appropriate degrees of freedom to decide whether to reject the null hypothesis.