AP - Statistics:: Inference for Categorical Data - Chi-Square

A CHI-SQUARE GOODNESS OF FIT TEST is used to determine whether:

- **A)** Two population means are equal
- **B)** A population proportion equals a specific value using a z-test
- **C)** A single categorical variable's distribution matches a HYPOTHESIZED (expected) distribution across its categories
- **D)** Two quantitative variables are correlated

The CHI-SQUARE TEST STATISTIC is calculated using the formula:

- **A)** \(\chi^2 = \sum \dfrac{(\text{Observed}-\text{Expected})^2}{\text{Expected}}\)
- **B)** \(\chi^2 = \text{Observed} \times \text{Expected}\)
- **C)** \(\chi^2 = \sum (\text{Observed}-\text{Expected})\)
- **D)** \(\chi^2 = \dfrac{\text{Observed}}{\text{Expected}}\)

In a chi-square goodness of fit test, EXPECTED COUNTS are calculated as:

- **A)** (Total sample size) \(\times\) (hypothesized proportion for that category)
- **B)** The total sample size divided by the observed count
- **C)** Always exactly equal for every category regardless of the hypothesis
- **D)** The observed count for that category

A die is rolled 60 times to test if it's fair (\(H_0\): each face has probability 1/6). What is the EXPECTED COUNT for each face?

- **A)** \(10\)
- **B)** \(6\)
- **C)** \(60\)
- **D)** \(1\)

The DEGREES OF FREEDOM for a chi-square goodness of fit test with \(k\) categories is:

- **A)** \(df = k\)
- **B)** \(df = n-1\)
- **C)** \(df = k-1\)
- **D)** \(df = k+1\)

For a chi-square goodness of fit test with 5 categories, find the degrees of freedom.

- **A)** \(5\)
- **B)** \(6\)
- **C)** \(1\)
- **D)** \(4\)

Which of the following is a required CONDITION for a valid chi-square test (of any type)?

- **A)** The sample size must be smaller than 10
- **B)** No conditions are needed for chi-square tests
- **C)** All expected counts must be exactly equal to each other
- **D)** RANDOM sample/assignment, INDEPENDENCE of observations, and Large Counts (all expected counts should generally be at least 5)

Why does the chi-square Large Counts condition check EXPECTED counts rather than OBSERVED counts?

- **A)** Expected counts are always larger than observed counts
- **B)** Observed and expected counts are always identical, making this distinction meaningless
- **C)** The chi-square distribution's approximation validity depends on the theoretical (expected) counts under the null hypothesis, since these represent what the sampling distribution theory assumes, regardless of what was actually observed
- **D)** This condition is checked using neither observed nor expected counts