DATASCI 6.3

What is the purpose of a chi-square goodness-of-fit test?

To determine whether observed categorical data matches an expected distribution

What type of data is required for a chi-square goodness of fit test?

Categorical (counts/ frequencies in categories)

State the null hypothesis for a goodness-of-fit test

The observed data follows the expected distribution

State the alternative hypothesis for a goodness-of-fit test

The observed data does not follow the expected distribution

What do O and E represent

O represents the observed counts and E represents the expected counts under H0

Why are the differences square in the chi-square formula

It prevents negatives from canceling positives and it emphasizes larger deviations

What does a large X^2 value indicate?

Strong evidence against the null hypothesis

What does a small X2 value indicate

Observed counts are closed to expected; supports the null hypothesis

Is the chi-square test one-tailed or two-tailed?

One-tailed (right-tailed only)

Why is the chi-square test right-tailed

Only large deviations, large X^2, provide evidence against H0

What does k represent in degrees of freedom

Number of categories (bins)

How does increasing degrees of freedom affect the chi-square distribution?

The distribution becomes more symmetric, center shifts right, and spread increases

What are the two main conditions for a chi-square test?

Independence and expected counts ≥ 5 in each category

Why must expected counts be at least 5?

To ensure the chi-square approximation is valid

What happens if expected counts are less than 5?

The test may not be reliable

What should you do if there are only two categories?

Use a one-proportion z-test instead

How is the chi-square statistic similar to a z-score?

It standardizes differences between observed and expected values, then combines them into one measure

What happens to X^2 if observed counts are very far from expected counts?

X^2 becomes large

How is the p-value found in a chi-square test?

Using the upper tail of the chi-square distribution with df= k-1

When do you reject the null hypothesis in a chi-square test?

When the p-value is less than the significance level

Why can't X^2 be negative?

Because all terms are squared

What does the chi-square statistic summarize?

The total deviation between observed and expected counts across all categories.