Which of the following accurately describes the chi-square test for goodness of fit?

• A. It is used to compare the means of two independent samples to see if they are significantly different.
• B. It is used to estimate the population variance from a small sample.
• C. It is used to test the linear relationship between two continuous variables.
• D. It is used to determine whether observed categorical data significantly differ from expected frequencies under a specific hypothesis.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the purpose of the chi-square test for goodness of fit. It is designed to compare observed categorical data with expected frequencies to see if there is a significant difference.
2. Step 2: Recognize that this test is not used for comparing means of samples, estimating variances, or testing linear relationships between continuous variables. Those are covered by other tests like t-tests, variance estimations, and correlation/regression analyses.
3. Step 3: Recall the formula for the chi-square statistic: \(\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\), where \(O_i\) is the observed frequency for category \(i\) and \(E_i\) is the expected frequency under the null hypothesis.
4. Step 4: The null hypothesis in a goodness of fit test states that the observed frequencies fit the expected distribution, while the alternative hypothesis states that they do not fit.
5. Step 5: By calculating the chi-square statistic and comparing it to a critical value from the chi-square distribution (based on degrees of freedom), we determine if the difference between observed and expected frequencies is statistically significant.