Unit 6: Inference for Categorical Data: Proportions

Categorical variable

A variable that places individuals into categories (e.g., yes/no, defective/not defective).

Success

The category of interest in a two-category (binary) setting; a label that does not imply “good.”

Failure

The other category in a two-category (binary) setting; a label that does not imply “bad.”

Population proportion (p)

The true fraction of the entire population that falls in the “success” category.

Sample proportion (p-hat)

The proportion of successes in a sample; computed as p̂ = x/n.

Inference

Using sample data to draw conclusions about a population parameter (here, a proportion).

Sampling variability

The natural variation in a statistic (like p̂) from one random sample to another, even when the true p is fixed.

Sampling distribution

The distribution of a statistic’s values over many repeated random samples from the same population.

Normal approximation for p-hat

For large enough samples, the sampling distribution of p̂ is approximately Normal (bell-shaped).

Association vs causation

Randomized experiments can support cause-and-effect conclusions; observational studies generally support only association due to possible confounding.

Number of successes (x)

The count of observations in the sample that fall in the “success” category.

Sample size (n)

The number of individuals/observations in the sample.

Null value (p0)

The claimed population proportion used in a hypothesis test (the value assumed under H0).

Parameter

A fixed (usually unknown) numerical characteristic of a population, such as p, p1, or p2.

Statistic

A numerical value computed from sample data that varies from sample to sample, such as p̂.

Independence assumption

Observations in a sample (and between groups, if applicable) must be independent for standard inference methods to apply.

Random sampling

Selecting individuals so each sample of a given size has an equal chance of being chosen; helps justify independence.

Random assignment

Assigning individuals to treatments by chance; helps create comparable groups and supports causal conclusions.

10% condition (10% Rule)

When sampling without replacement, independence is reasonable if n ≤ 0.10N, where N is population size.

Normality assumption (for proportions)

Using a Normal model for the sampling distribution of p̂; justified when expected success and failure counts are large enough.

q = 1 − p

The population proportion of failures (the complement of the success proportion p).

Large counts condition

A check that expected numbers of successes and failures are at least 10, supporting the Normal approximation.

Binomial model

A model for the number of successes x when trials are independent and the success probability is constant: x ~ Binomial(n, p).

Unbiased estimator

An estimator whose expected value equals the true parameter; E(p̂) = p under random sampling.

Standard deviation of p-hat (σ_p̂)

The true SD of the sampling distribution of p̂: σ_p̂ = sqrt(p(1−p)/n).

Standard error (SE)

An estimate of a sampling distribution’s standard deviation, computed using sample data or the null value.

SE for one-proportion confidence interval

Estimated SD of p̂ for a CI: SE = sqrt(p̂(1−p̂)/n).

SE under the null (one-proportion test)

SD used in a one-proportion z test, computed with p0: sqrt(p0(1−p0)/n).

One-proportion z confidence interval

An interval estimating p: p̂ ± z* sqrt(p̂(1−p̂)/n).

Critical value (z*)

The z-score multiplier that matches a chosen confidence level (e.g., about 1.96 for 95%).

Margin of error (ME)

The “plus/minus” amount in a confidence interval: ME = z* × SE.

Confidence level

The long-run success rate of the CI method (e.g., 95% of such intervals capture the true parameter).

Correct confidence interval interpretation

A statement about being C% confident that the true population proportion (parameter) lies between the interval bounds, in context.

Significance test

A procedure that assesses whether data provide convincing evidence against a specific null claim about a parameter.

Null hypothesis (H0)

The claim assumed true for the test, stated as an equality for proportions (e.g., H0: p = p0).

Alternative hypothesis (Ha)

The competing claim, stated as an inequality (p ≠ p0, p > p0, or p < p0), chosen from the question context.

One-proportion z test statistic

A standardized measure for testing p: z = (p̂ − p0) / sqrt(p0(1−p0)/n).

p-value

Assuming H0 is true, the probability of getting a result at least as extreme as the observed statistic (in the direction of Ha).

Significance level (α)

A chosen cutoff for deciding whether a p-value is “small” (often 0.05); also the probability of a Type I error.

Type I error

Rejecting H0 when H0 is actually true; its probability is controlled by α.

Type II error

Failing to reject H0 when Ha is actually true.

Beta (β)

The probability of a Type II error for a specific alternative value of the parameter.

Power

1 − β; the probability of rejecting a false H0 when a particular alternative is true.

Two independent samples design

A design with separate random samples from two populations; used to compare p1 and p2.

Randomized experiment design

A design where subjects are randomly assigned to treatments; differences in outcomes can be attributed to the treatment (in that setting).

Two population proportions (p1, p2)

p1 and p2 are the true success proportions in populations or treatment groups 1 and 2.

Difference in sample proportions (p̂1 − p̂2)

A statistic that estimates the parameter p1 − p2; computed from two samples/groups.

Unpooled SE for two-proportion confidence interval

SE for a CI for p1 − p2: sqrt( p̂1(1−p̂1)/n1 + p̂2(1−p̂2)/n2 ).

Pooled proportion (p̂c)

Combined estimate used in two-proportion tests under H0: p̂c = (x1 + x2)/(n1 + n2).

Two-proportion z test statistic

For H0: p1 = p2, z = ((p̂1 − p̂2) − 0) / sqrt( p̂c(1−p̂c)(1/n1 + 1/n2) ).