Selecting Appropriate Inference Procedures: Categorical Data - AP Statistics

What is required for a valid chi-square test of homogeneity?

Random samples from each group. Ensures valid comparison across independent groups.
What are expected frequencies in a chi-square test?

Theoretical frequencies if null hypothesis is true. Calculated assuming the null hypothesis is true.
Which test checks if proportions differ across groups?

Chi-square test of homogeneity. Compares proportions of a categorical variable across multiple groups.
What statistic measures how well observed frequencies fit expected frequencies?

Chi-square statistic. Measures the discrepancy between observed and expected values.
Identify the test: Comparing observed vs. expected frequencies in categories.

Chi-square goodness-of-fit test. Tests if observed data matches a theoretical distribution.
Identify the test used for paired categorical data.

McNemar's test. Analyzes matched pairs with binary outcomes.
Which test is used for exact tests of independence in small samples?

Fisher's exact test. Computes exact probabilities for small sample contingency tables.
Identify the degrees of freedom for a chi-square test of independence.

$(r-1)(c-1)$ where $r$ is rows, $c$ is columns. Calculated using rows minus 1 times columns minus 1.
What conclusion is drawn if $\chi^2$ is significant in a test of independence?

Variables are not independent. Evidence of association between the categorical variables.
What outcome does a large chi-square statistic suggest?

Greater deviation from the expected frequencies. Indicates poor fit between observed and expected data.
What does a significant chi-square statistic indicate?

Observed frequencies differ from expected frequencies. Evidence against the null hypothesis of independence or fit.
What is a contingency table used for in statistics?

To display the frequency distribution of variables. Shows the relationship between two categorical variables.
When is a chi-square test of homogeneity used?

To compare distributions of a categorical variable across groups. Examines whether groups have the same categorical distribution.
What assumption is crucial for chi-square tests?

Expected frequencies should be at least 5. Ensures valid chi-square distribution approximation.
What is the null hypothesis for a chi-square test of independence?

The variables are independent. This states no relationship exists between the variables.
Which method can be used when chi-square assumptions are violated?

Fisher's exact test. Provides exact probabilities without large-sample assumptions.
Identify the test: Comparing categorical data from more than two groups.

Chi-square test of homogeneity. Tests if distributions are the same across different populations.
Identify the test: Testing a single categorical distribution.

Chi-square goodness-of-fit test. Examines if data follows a specific categorical distribution.
Which test is appropriate for comparing two proportions?

Two-sample z-test for proportions. Used when comparing proportions from two independent groups.
What is the purpose of Fisher's exact test?

To assess the significance of the association in a $2\times^2$ table. Used when sample sizes are too small for chi-square test.
Which test examines whether distribution matches a specified distribution?

Chi-square goodness-of-fit test. Compares one sample distribution to a theoretical model.
Identify the type of data suitable for McNemar's test.

Paired categorical data. Uses matched pairs from the same subjects.
What is the null hypothesis for a chi-square test of homogeneity?

Proportions are the same across groups. Assumes no difference in distributions between groups.
Which inference procedure is appropriate for categorical data in one group?

Chi-square goodness-of-fit test. Tests distribution shape for a single categorical variable.
What is the alternative hypothesis for a chi-square test of independence?

The variables are not independent. This states a relationship exists between the variables.
For what type of data is a chi-square test inappropriate?

Continuous data. Chi-square tests require categorical, not continuous variables.
How is the chi-square distribution described?

A family of distributions that vary with degrees of freedom. Shape depends on degrees of freedom parameter.
Find the degrees of freedom: $3\times^4$ contingency table.

Degrees of freedom = $(3-1)(4-1) = 6$ . Multiply rows minus 1 by columns minus 1.
State the formula for the chi-square statistic.

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ . Sums squared differences between observed and expected, divided by expected.
Which test would you use for a $2\times^2$ contingency table with small counts?

Fisher's exact test. Appropriate when chi-square assumptions aren't met.
What does a p-value in a chi-square test indicate?

Probability of observing data as extreme as current, under null hypothesis. Probability of obtaining results this extreme by chance.
Which test is used for two related samples in categorical data?

McNemar's test. Compares before and after measurements on same subjects.
When should Fisher's exact test be used instead of chi-square?

When sample sizes are too small for chi-square assumptions. When expected cell counts fall below 5.
What is the null hypothesis for a chi-square goodness-of-fit test?

Sample distribution fits the expected distribution. Assumes observed data matches the theoretical model.
Which test would you use for the relationship between two nominal variables?

Chi-square test of independence. Tests association between two categorical variables.
What is the primary use of a chi-square test?

To test categorical data for independence or fit. Analyzes relationships and distributions in categorical data.
What is the key assumption of the chi-square test of independence?

Observations are independent. Each observation must not influence others.

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