Math 183 - Part 6 (Probability Distributions)

Representative sample

A sample that accurately reflects the population, necessary for valid inference.

Proportion Parameter (p*)

The true proportion of a population with a certain characteristic (e.g., support for a candidate).

Sample Proportion (p̂)

The observed proportion of a sample with a given characteristic

Bernoulli random variable

A variable that takes value 1 for “success” and 0 for “failure”; used to model binary responses.

Binomial distribution (in surveys)

Models the number of “successes” in n independent binary responses.

Confidence interval for a proportion

A range likely to contain the true population proportion

Pooled Proportion

A combined estimate of proportion across two samples

Confidence interval for difference in proportions

A range for the difference

Hypothesis test for two proportions

Test statistic where p̂ is the pooled proportion

Bonferroni correction

Adjusts confidence level or significance when making multiple comparisons to control overall error rate.

Simultaneous confidence intervals

Multiple CIs adjusted (e.g., via Bonferroni) so that all intervals together have the correct confidence level.

Pearson’s chi-squared test (homogeneity)

Compares observed vs expected counts to test if proportions are the same across groups.

Chi-squared test statistic

where O = observed count and E = expected count.

Degrees of freedom for chi-squared

For J groups and K categories: (J - 1)(K - 1)

Contingency table

A table showing counts for combinations of two categorical variables.

Independence of categorical variables

Two variables are independent if the joint probability equals the product of their marginal probabilities.

Chi-squared test for independence

Tests whether two categorical variables are independent using observed and expected counts in a contingency table.

P-value in chi-squared test

The probability of observing a test statistic as extreme as (or more than) what was seen, assuming the null hypothesis.

Interpretation of a significant chi-squared

If p-value < α, conclude there is evidence that proportions (or association) differ.

Simultaneous inference

Making multiple comparisons or constructing multiple intervals while adjusting to maintain correct overall error rates.

Pooled estimation (in tests)

Use the combined estimate of a parameter when the null hypothesis assumes no difference.

Separate estimation (in CIs)

Estimate parameters separately for each group when constructing confidence intervals.