Expected counts are the values we anticipate seeing in each category if the null hypothesis is true.

The chi-square statistic quantifies how much our observed data deviates from our expected counts.

Degrees of freedom (df) is a parameter that determines the shape of the chi-square distribution; calculated as (Number of Categories) - 1.

The chi-square goodness of fit (GOF) test checks if the observed frequencies of a categorical variable match a hypothesized distribution.

The null hypothesis is the assumption we're testing – usually, it states there's no difference or relationship between variables.

The p-value tells us the probability of getting our observed results (or more extreme) if the null hypothesis were true. A small p-value (typically < 0.05) means our results are statistically significant, and we reject the null hypothesis.

Chi-square distributions are always positive and skewed to the right. The shape is determined by the degrees of freedom; as df increases, the distribution becomes more symmetrical.

The Large Counts condition requires that all expected counts must be at least 5. This ensures the chi-square statistic is approximately chi-square distributed.

The sample must be randomly selected to ensure that the sample is representative of the population, which is a requirement for the GOF test to be valid.

The population must be at least 10 times the sample size (10% rule). This ensures that the observations are independent of each other.

Expected Count = (Sample Size) x (Probability under Null Hypothesis)

\( \chi^2 = \sum \frac{(Observed - Expected)^2}{Expected} \)

df = (Number of Categories) - 1

Expected Count = (Total Sample Size) * (Proportion under Null Hypothesis). E.g., for Harry Potter: 2500 * 0.33 = 825

\( \chi^2 = \frac{(180-200)^2}{200} + \frac{(150-150)^2}{150} + \frac{(170-150)^2}{150} \)