AP Stats chi-square test for independence practice questions

Chi-square test purpose?

To determine if there is a significant association between two categorical variables.

True or False: Chi-square tests can only be used with normal distributions.

False - Chi-square tests are used for categorical data, not dependent on normality.

Degrees of freedom formula?

Calculated as: (rows - 1) × (columns - 1).

What is a contingency table?

A table displaying the frequency distribution of variables. Example: Survey responses vs. age groups.

Expected frequency calculation?

For cell (i, j): $\displaystyle E_{ij} = \\frac{(row\ total)(column\ total)}{grand\ total}$.

True or False: Larger chi-square values indicate stronger associations.

True - A larger chi-square value suggests a greater deviation from independence.

What data type does chi-square test use?

Categorical data, such as gender, favorite color, or product type.

Rejecting the null hypothesis means?

There is significant evidence of an association between the variables being tested.

Example of chi-square test?

A store tests if customer preferences differ by age group. Collect data and analyze.

Cause → Effect: High chi-square value.

Indicates a significant association between the two categorical variables.

What assumptions must be met?

Observations must be independent; expected frequency should be at least 5.

Fill in the blank: Chi-square tests compare _____ and expected frequencies.

observed frequencies.

Comparison: Chi-square test vs. t-test.

Chi-square tests assess categorical variables; t-tests assess means of continuous variables.

P-value interpretation?

A low p-value (< 0.05) indicates strong evidence against the null hypothesis.

What does the null hypothesis state?

There is no association between the two categorical variables.

True or False: Chi-square tests can be used for small sample sizes.

False - Small samples may lead to unreliable results in chi-square tests.

A store sells 3 types of shoes: sneakers, boots, sandals.

If the store surveys 200 customers, and finds 80 prefer sneakers, 70 prefer boots, and 50 prefer sandals, conduct a chi-square test to see if preference is independent of shoe type.

True or False: Chi-square tests can only be used for categorical data.

True - Chi-square tests are designed to analyze categorical data to assess independence.

If a restaurant offers vegetarian and non-vegetarian options, how to test preference?

Survey 100 customers: 40 choose vegetarian, 60 choose non-vegetarian. Use the chi-square test to evaluate if the choice is independent of the type of option.

Fill in the blank: The chi-square test compares ____ and ____ frequencies.

observed; expected

A city analyses bike usage by gender: male, female. Data shows 60 males, 30 females use bikes.

Perform a chi-square test to determine if gender and bike usage are independent from each other.

Example: A survey shows 40% like coffee, 30% tea, and 30% neither.

Test if preference for drinks is independent of age group (under 30, over 30) using chi-square.

A school wants to know if students' grades affect club participation.

Survey shows 30 A's, 40 B's, 30 C's in clubs. Use chi-square test to analyze independence.

True or False: A significant chi-square result indicates no association.

False - A significant result indicates an association exists between the variables.

Compare: Chi-square test vs. t-test.

Chi-square tests for independence (categorical data) while t-tests compare means (numerical data).

A survey of 500 people on movie preference produces: Action 200, Comedy 150, Drama 150.

Investigate if age (under 25, 25+) impacts these preferences using a chi-square test.

Determine: Are sports preferences independent of gender?

Survey: 50 males prefer basketball, 30 females prefer soccer. Conduct chi-square test.

When is it appropriate to use a chi-square test?

When analyzing the relationship between two categorical variables based on frequency counts.

What do you conclude if chi-square p-value < 0.05?

Reject the null hypothesis; there is a significant association between the two categorical variables.

Fill in the blank: The null hypothesis for chi-square tests states that variables are _____.

independent

A pet store tracks sales of cats vs dogs across different months.

Analyze the data to see if sales patterns are independent across seasons using a chi-square test.

A restaurant analyzes dining preferences by age group. What to do?

Conduct a chi-square test for independence. Data might show preferences for dining in or takeout among age groups. If p-value < 0.05, conclude age affects dining preference.

What does a p-value indicate in a chi-square test?

The p-value assesses the strength of evidence against the null hypothesis. A low p-value (typically < 0.05) suggests rejecting the null hypothesis.

True or False: A high chi-square statistic guarantees a significant result.

False. A high chi-square statistic indicates a large difference between observed and expected frequencies, but significance is determined by the p-value.

If the p-value is 0.03, what conclusion can be drawn?

Since 0.03 < 0.05, reject the null hypothesis. There is significant evidence of a relationship between the variables.

Interpret a chi-square statistic of 10.5 with df = 4.

With df = 4, a statistic of 10.5 corresponds to a p-value below 0.05, suggesting a significant relationship between variables.

Fill in the blank: To establish independence, we need a p-value that is ____.

greater than 0.05.

Cause → Effect: What does a significant result imply?

It implies a relationship exists between the two categorical variables being tested.

Comparison: Chi-square test vs. Fisher's Exact Test

Chi-square is used for larger samples, while Fisher's is used for smaller samples or when expected frequencies are low.

How do you interpret a chi-square test result of not significant?

Fail to reject the null hypothesis; conclude no evidence of an association between the variables.

What is the critical value of chi-square for df = 3 at 0.05 significance?

The critical value is approximately 7.815. If the chi-square statistic exceeds this, it's significant.

True or False: The chi-square test can be used for continuous data.

False. It is only applicable to categorical data.

Interpretation of expected frequencies in a chi-square test.

Expected frequencies are the counts we would expect if the null hypothesis were true, calculated from marginal totals.

A chi-square test shows a large discrepancy in observed vs expected values. What now?

Investigate the association further; consider potential confounding variables or larger sample sizes.