Unit 3 · Inference for Categorical Data: Proportions

Estimator ● (core concept) — A sample statistic used to estimate a population parameter.

Unbiased estimator ● (core concept) — An estimator that, on average, neither underestimates nor overestimates the population parameter.

Point estimator ● (core concept) — A sample statistic regarded as the single-value estimate of the corresponding population parameter; for example, the sample proportion p̂ is a point estimator for the population proportion p.

Sampling distribution of a sample proportion ● (core concept) — The distribution of sample-proportion values over all possible samples of size n. When the sampled values are independent, its mean is μ(p̂) = p and its standard deviation is σ(p̂) = √(p(1−p)/n).

Randomization condition (proportions) ● (core concept) — The data should be collected using a random sample — or, for two-sample procedures, two independent random samples or a randomized experiment. If the data come from an experiment, only this condition is needed, with treatments randomly assigned to experimental units.

10% condition (proportions) ● (core concept) — When sampling without replacement, the population size N must be at least 10 times the sample size (n ≤ 10%N); for two samples, n1 ≤ 10%N1 and n2 ≤ 10%N2. This condition is unnecessary when the data are from a randomized experiment.

Large counts condition (sampling distribution of a sample proportion) ● (core concept) — The sampling distribution of p̂ is approximately normal provided np ≥ 10 and n(1−p) ≥ 10, where np is the expected number of successes and n(1−p) the expected number of failures.

One-sample z-interval for a population proportion ● (core concept) — The confidence interval procedure for a single population proportion: p̂ ± z*√(p̂(1−p̂)/n).

Confidence interval ● (core concept) — An interval estimate for a population parameter, constructed as point estimate ± margin of error.

Stating the parameter in context (proportion interval) ● (core concept) — The population value being estimated; for a proportion confidence interval it should reference the proportion, the response variable, and the population in context.

Confidence level ● (core concept) — The C% attached to a confidence interval: in repeated random sampling with the same sample size, approximately C% of the confidence intervals calculated will capture the true population parameter.

Critical value (z*) ● (core concept) — z* (and −z*) are the values enclosing the middle C% of the standard normal distribution, where C% is the approximate confidence level.

Standard error (of a statistic) ● (core concept) — An estimate of the standard deviation of the sampling distribution of the statistic; it quantifies the typical amount a statistic will vary from the corresponding population parameter. For a sample proportion, SE(p̂) = √(p̂(1−p̂)/n).

Margin of error ● (core concept) — Half the width of the confidence interval: the critical value times the standard error (for a proportion, z*√(p̂(1−p̂)/n)). Rearranged, it gives the minimum sample size n = (z*²·p̂(1−p̂))/MOE², using p̂ = 0.5 when p̂ is unknown to get an upper bound for n.

Point estimate ● (core concept) — The single value used to estimate the parameter: p̂ for a population proportion, x̄ for a population mean, p̂1 − p̂2 for a difference in proportions, and x̄1 − x̄2 for a difference in means.

Normality condition (one-sample z-interval for a population proportion) ● (core concept) — The observed number of successes, n·p̂, and the observed number of failures, n·(1−p̂), should each be at least 10.

Confidence interval interpretation (population proportion) ● (core concept) — We are C% confident that the interval (a, b) contains the true value of the population proportion, where a and b are the lower and upper limits. Because the interval is calculated from a sample, it may or may not contain the parameter.

Relationships among confidence level, margin of error, sample size, and interval width ● (core concept) — For a given sample, increasing the confidence level increases the critical value, the margin of error, and the width of the interval. Increasing the sample size decreases the standard error and narrows the interval (width is approximately proportional to 1/√n).

Justifying a claim based on a confidence interval (population proportion) ● (core concept) — A confidence interval for a population proportion provides a range of plausible values that may serve as convincing evidence to support a particular claim about the population proportion.

One-sample z-test for a population proportion ● (core concept) — The hypothesis testing procedure for a population proportion, used to make a decision about the value of the parameter.

Hypothesis test ● (core concept) — A statistical inference procedure used to make a decision about the value of a population parameter.

Null hypothesis ● (core concept) — H0, the statement about a parameter assumed to be correct unless convincing statistical evidence suggests otherwise; it is the status quo. It contains an equality reference (=, ≤, or ≥), and for a one-sided test it is tested at the boundary of equality. For a one-sample proportion test, H0: p = p0, where p0 is the null hypothesized value.

Alternative hypothesis ● (core concept) — Ha, the claim or belief about a parameter for which evidence is being collected; it represents the researcher's claim. An alternative with < or > is one-sided; with ≠ it is two-sided. For a one-sample proportion test, Ha: p < p0, Ha: p > p0, or Ha: p ≠ p0.

Normality condition (one-sample z-test for a population proportion) ● (core concept) — The expected number of successes, np0, and the expected number of failures, n(1−p0), should each be at least 10.

Stating the parameter in context (proportion test) ● (core concept) — For a hypothesis test for a population proportion, the parameter should reference the population parameter, the response variable, and the population in context.

p-value ● (core concept) — Assuming the null hypothesis is true, the probability of obtaining a test statistic as extreme or more extreme — in the direction of the alternative hypothesis — than the observed test statistic. For Ha: > it is the probability at or above the observed statistic; for Ha: < the probability at or below it; for Ha: ≠ the sum of the tail probabilities at −|observed| and +|observed|. Small p-values provide convincing evidence for Ha; p-values that are not small provide no convincing evidence for Ha — nor evidence that H0 is true.

Null distribution ● (core concept) — Given that the null hypothesis is true, the probability distribution of the test statistic, from which the p-value is found. If the null distribution has been simulated, the p-value is the proportion of simulated values as extreme or more extreme than the observed test statistic.

Test statistic (one-sample z-test for a proportion) ● (core concept) — z = (p̂ − p0)/√(p0(1−p0)/n). The z-statistic has a standard normal distribution when the null hypothesis is true, and its p-value is found from the standard normal distribution using a table or technology.

Significance level ● (core concept) — Denoted α, the predetermined probability of rejecting the null hypothesis given that it is true; it may be given or determined by the researcher.

Statistical significance ● (core concept) — Determined by the relationship between the p-value and the significance level of the hypothesis test.

Decision rule (hypothesis test) ● (core concept) — Explicitly compare the p-value to the significance level α: if the p-value ≤ α, reject the null hypothesis; if the p-value > α, fail to reject it. Rejecting means there is convincing statistical evidence for the alternative; failing to reject means there is not — a test can never prove the null hypothesis true.

Hypothesis test conclusion ● (core concept) — Stated in context, consistent with and in terms of the alternative hypothesis, using non-definitive language, with a reference to the parameter and the population.

Type I error ● (core concept) — Concluding there is convincing statistical evidence that the alternative hypothesis is true (due to a small p-value) when it is not. Its probability is defined as the significance level α, typically set to a small value (e.g., 0.01, 0.05, 0.10) before data are collected.

Type II error ● (core concept) — Failing to find convincing statistical evidence that the alternative hypothesis is true (due to a large p-value) when it is true. Its probability equals 1 − power.

Power ● (core concept) — The probability that a hypothesis test correctly rejects a false null hypothesis; ideally large (e.g., 0.80).

Factors that increase power ● (core concept) — For a given study and test, the probability of a Type II error decreases and power increases when any of these change (others held constant): sample size increases, standard error decreases, the true parameter value is farther from the null, or the significance level increases.

Consequences of Type I and Type II errors ● (core concept) — The seriousness of each error type should be weighed before the study: consequences of a Type I error influence the choice of significance level, and consequences of a Type II error influence how large the sample size should be.

Sampling distribution of the difference in sample proportions ● (core concept) — For two independent populations with proportions p1 and p2, when sampled values are independent, the sampling distribution of p̂1 − p̂2 has mean p1 − p2 and standard deviation √(p1(1−p1)/n1 + p2(1−p2)/n2).

Large counts condition (sampling distribution of the difference in sample proportions) ● (core concept) — The sampling distribution of p̂1 − p̂2 is approximately normal provided n1p1 ≥ 10, n1(1−p1) ≥ 10, n2p2 ≥ 10, and n2(1−p2) ≥ 10, where these are the expected numbers of successes and failures in each sample.

Two-sample z-interval for a difference between population proportions ● (core concept) — The confidence interval procedure for the difference between two population proportions: (p̂1 − p̂2) ± z*√(p̂1(1−p̂1)/n1 + p̂2(1−p̂2)/n2).

Standard error (difference in proportions) ● (core concept) — SE(p̂1 − p̂2) = √(p̂1(1−p̂1)/n1 + p̂2(1−p̂2)/n2).

Normality condition (two-sample z-interval for a difference in proportions) ● (core concept) — The observed numbers of successes (n1·p̂1, n2·p̂2) and failures (n1(1−p̂1), n2(1−p̂2)) for both samples must all be at least 10.

Stating the parameters in context (two-sample proportions) ● (core concept) — For a two-sample z-interval for the difference between two population proportions, the parameters should refer to the difference in the proportions, the response variable, and the populations in context; for a two-sample z-test, the parameters should reference the population parameters, the response variables, and the populations in context.

Confidence interval for a difference in proportions (interpretation and claim) ● (core concept) — We are C% confident the interval (a, b) contains the true difference between the two population proportions; the computed interval may or may not contain it. If the interval contains 0, there is insufficient evidence of a difference; if it does not contain 0, there is sufficient evidence of a difference.

Two-sample z-test for a difference between two population proportions ● (core concept) — The hypothesis testing procedure for comparing two population proportions.

Pooled proportion ● (core concept) — p̂c = (n1·p̂1 + n2·p̂2)/(n1 + n2), the combined proportion of successes for the two groups, used as the common proportion assuming H0 is true.

Null and alternative hypotheses (two-sample z-test for proportions) ● (core concept) — H0 states no difference: H0: p1 = p2 or H0: p1 − p2 = 0. One-sided alternatives are Ha: p1 < p2 (or p1 − p2 < 0) or Ha: p1 > p2 (or p1 − p2 > 0); the two-sided alternative is Ha: p1 ≠ p2 (or p1 − p2 ≠ 0).

Normality condition (two-sample z-test for a difference in proportions) ● (core concept) — n1·p̂c, n1(1−p̂c), n2·p̂c, and n2(1−p̂c) must all be at least 10, where p̂c is the pooled proportion.

Test statistic (two-sample z-test for proportions) ● (core concept) — z = ((p̂1 − p̂2) − 0)/√(p̂c(1−p̂c)(1/n1 + 1/n2)), using the pooled proportion p̂c. The z-statistic has a standard normal distribution when the null hypothesis is true.

Chi-square statistic ● (core concept) — Measures the distance between observed and expected counts relative to expected counts.

Chi-square distribution ● (core concept) — A family of density curves with only positive values that are skewed right; the skew becomes less pronounced as the degrees of freedom increase.

Chi-square test for homogeneity ● (core concept) — Used to determine whether the distribution of a categorical variable differs across two or more populations (or treatments); it should reference the categorical variable and the populations in context.

Chi-square test for independence ● (core concept) — Used to determine whether two categorical variables in a two-way table might be associated in the single population sampled; it should reference the categorical variables and the population in context.

Null and alternative hypotheses (chi-square test for homogeneity) ● (core concept) — H0: there is no difference in the distribution of the categorical variable across populations or treatments. Ha: there is a difference in the distribution of the categorical variable across populations or treatments.

Null and alternative hypotheses (chi-square test for independence) ● (core concept) — H0: there is no association between the two categorical variables (they are independent) in the given population. Ha: there is an association between the two categorical variables (they are not independent) in the given population.

Randomization condition (chi-square test) ● (core concept) — For the test of independence, the data should be collected using a random sample; for the test of homogeneity, the data should be collected using independent random samples or a randomized experiment.

10% condition (chi-square test) ● (core concept) — When sampling without replacement, check that n ≤ 10%N, where N is the population size and n the sample size. This condition is unnecessary when the data are from a randomized experiment.

Expected counts condition (chi-square test) ● (core concept) — All expected counts should be greater than 5.

Expected counts (two-way table) ● (core concept) — Under the null hypothesis, the expected count in a cell equals (row total × column total) / table total.

Chi-square test statistic ● (core concept) — χ² = Σ((Observed count − Expected count)² / Expected count), summed over all cells of the two-way table.

Degrees of freedom (chi-square) ● (core concept) — df = (number of rows − 1)(number of columns − 1); the chi-square statistic follows this chi-square distribution when the null hypothesis is true.

p-value for a chi-square test ● (core concept) — The p-value for a chi-square test for independence or homogeneity is found from a chi-square distribution using a table or technology.