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

The observed distribution does not match the expected distribution. This states a significant difference exists between observed and expected distributions.

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$. Sums squared differences between observed and expected, divided by expected.

$O_i$ represents the observed frequency for category $i$. This is the actual count observed in each category of the data.

$E_i$ represents the expected frequency for category $i$. This is the theoretical count predicted for each category under $H_0$.

Degrees of freedom = number of categories - 1. Subtract 1 from categories because we lose one degree of freedom.

A common significance level is $\alpha = 0.05$. This is the standard threshold for statistical significance in most tests.

The p-value is the probability of observing a test statistic as extreme as the one observed. Measures how likely the observed result is if the null hypothesis is true.

Reject the null hypothesis. When p-value < $\alpha$, evidence strongly contradicts the null hypothesis.

Fail to reject the null hypothesis. When p-value ≥ $\alpha$, insufficient evidence exists to reject the null hypothesis.

Chi-Square Goodness of Fit test. Tests if observed roll frequencies match expected equal probabilities for fairness.

Expected frequency = 50 for each category. Divide total trials (200) by number of categories (4) for equal distribution.

Chi-Square distribution. The test statistic follows this right-skewed distribution under the null hypothesis.

Expected frequency in each category should be at least 5. This ensures the normal approximation to the Chi-Square distribution is valid.

A large deviation between observed and expected frequencies. Large values suggest the observed data doesn't fit the expected model well.

A small deviation between observed and expected frequencies. Small values suggest the observed data closely matches the expected model.

To test how well an observed distribution fits an expected distribution. This test determines if data follows a specific theoretical distribution pattern.

Chi-Square Goodness of Fit test. This test compares actual observations against theoretical expectations or models.

Degrees of freedom = 4. Apply the formula: df = categories - 1 = 5 - 1 = 4.

Observed frequencies match expected frequencies exactly. Zero indicates perfect agreement between observed and expected frequency values.

To find p-values and critical values for Chi-Square tests. The table provides probability values for different Chi-Square statistics and df.

Chi-Square Goodness of Fit test. Tests if observed heads/tails frequencies match expected 50/50 distribution.

$\frac{(30 - 25)^2}{25} = 1.0$. Apply the formula: $(30-25)^2/25 = 25/25 = 1.0$.

Whether observed frequencies significantly differ from expected frequencies. Determines if the sample data matches a proposed theoretical distribution model.

Random sampling and independence of observations. These ensure valid probability calculations and prevent bias in the test.

There is a statistically significant difference; reject the null hypothesis. The difference is statistically significant at the 5% level of significance.

Skewed to the right, especially with low degrees of freedom. The distribution becomes more symmetric as degrees of freedom increase.

Mean = $k$. The mean equals the number of degrees of freedom in the distribution.

Variance = $2k$. The variance is always twice the degrees of freedom value.

To test if observed data fits a specified distribution. Determines if sample data follows a hypothesized probability distribution pattern.