Chi-Square Goodness of Fit (Test) - AP Statistics

Greater evidence against the null hypothesis. Higher values indicate greater deviation from expected.

It becomes more symmetric. Approaches normal distribution as degrees of freedom increase.

7.815. From chi-square table with $\alpha = 0.05$ and $df = 3$.

$df = k - 1$. Where $k$ is the number of categories being tested.

P-value indicates the probability of observing a test statistic as extreme. Area under chi-square curve beyond observed statistic.

Strong evidence against the null hypothesis. Small p-values suggest data unlikely under null hypothesis.

Sample size should be large enough to ensure expected frequencies are at least 5. Prevents invalid approximation to chi-square distribution.

Higher Chi-Square statistic typically results in a lower p-value. Larger deviations produce higher statistics and lower p-values.

Chi-Square test may not be valid. Violates assumptions needed for valid statistical inference.

Chi-Square Goodness of Fit test. Tests if sample matches theoretical distribution pattern.

Null hypothesis. Reject when evidence strongly contradicts null assumption.

To determine if observed frequencies match expected frequencies. Tests if data follows a specific distribution pattern.

Chi-Square statistic > critical value. Decision rule for hypothesis testing with chi-square.

Fail to reject the null hypothesis. Insufficient evidence to reject null hypothesis.

To compare against observed frequencies. Baseline for measuring deviation from null hypothesis.

Data must be categorical. Chi-square tests require frequency counts of categories.

0.05. Most commonly used threshold for statistical significance.

Degrees of freedom = 9. Formula: $df = k - 1$ where $k$ is categories.

Increases the power of the test. Larger samples better detect true differences from null.

$O_i$ represents the observed frequency. The subscript $i$ denotes each category in the data set.

Expected frequency should be at least 5. Ensures valid approximation to chi-square distribution.

$E_i$ represents the expected frequency. The subscript $i$ denotes each category under null hypothesis.

Degrees of freedom = 4. Formula: $df = k - 1$ where $k$ is number of categories.

The observed frequencies match the expected frequencies. Assumes the theoretical distribution is correct.

The observed frequencies do not match the expected frequencies. At least one category differs from expected distribution.

To determine whether to reject the null hypothesis. Compare with test statistic to make statistical decision.

$E_i = \frac{\text{Total Observed} \times \text{Proportion}}{1}$. Multiply total sample size by theoretical probability.

Chi-Square distribution table. Provides critical values based on degrees of freedom.

Goodness of Fit and Test of Independence. Both test distribution assumptions but in different ways.