Inference for Categorical Data: Proportions

Fifteen topics on the move from a sample to a claim about a population, done for categorical data. Estimators and the sampling distribution that makes them trustworthy, confidence intervals for one proportion and for the difference between two, what a confidence level is actually a property of, significance tests and what a p-value is and is not, the two kinds of error and what each one costs, and chi-square tests for two or more categories. This unit is the hinge of the course: the procedures here repeat, with new arithmetic, in Unit 4.

Exam weight 15-25%15 topics

Topics

3.1

Estimators

Sampling Distributions for Sample Proportions

Constructing a Confidence Interval for a Population Proportion

Justifying a Claim Based on a Confidence Interval for a Population Proportion

Setting Up a Test for a Population Proportion

p-Values

Carrying Out a Test for a Population Proportion

Potential Errors When Performing Tests

Sampling Distributions for the Difference Between Sample Proportions

Constructing a Confidence Interval for the Difference Between Two Population Proportions

Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions

Setting Up a Test for the Difference Between Two Population Proportions

Carrying Out a Test for the Difference Between Two Population Proportions

Setting Up a Chi-Square Test for Homogeneity or Independence

Carrying Out a Chi-Square Test for Homogeneity or Independence

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Statistic vs parameter

p̂ is computed from the sample; p is the unknown being estimated

Unbiased estimator

its sampling distribution is CENTERED at the parameter, not "right every time"

Spread of p̂

√(p(1 − p) / n). Quadrupling n halves it

Three conditions

randomization; 10% (sampling without replacement); large counts

Large counts, INTERVAL

n·p̂ and n(1 − p̂) both at least 10

One-proportion interval

p̂ ± z* · √(p̂(1 − p̂) / n)

Two-proportion interval

(p̂₁ − p̂₂) ± z* · √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂). NOT pooled

Interpreting an interval

plausible values for the PARAMETER, in context. Not for the data, not for future samples

Interpreting the LEVEL

a property of the METHOD over many samples: about 95% of such intervals capture p

Judging a claim

values inside are plausible; a claim entirely outside is contradicted by the data

Hypotheses

always about the PARAMETER, stated before the data: H₀: p = p₀ against a one- or two-sided Hₑ

One-proportion z

z = (p̂ − p₀) / √(p₀(1 − p₀) / n): the null value p₀ goes in the standard error, not p̂

Large counts, TEST

n·p₀ and n(1 − p₀) both at least 10

p-value

P(a statistic at least this extreme GIVEN H₀ is true). Not P(H₀ true), not "the chance it was random"

Decision

p-value ≤ α: reject H₀. Otherwise FAIL TO REJECT: never "accept", never "prove"

Two-proportion test

the null says the proportions are equal, so POOL: p̂_c = (x₁ + x₂) / (n₁ + n₂)

Type I error

rejecting a TRUE null. Its probability is α

Type II error

failing to reject a FALSE null. Its probability is β

Power

1 − β: the chance of detecting a real effect. Rises with n, with α, and with a bigger true effect

Chi-square statistic

Σ (observed − expected)² / expected, over every cell

Expected count

(row total × column total) / grand total. Condition: every EXPECTED count at least 5

Chi-square df

(rows − 1)(columns − 1). Never n − 1

Homogeneity vs independence

several populations compared on one variable, versus ONE population classified two ways

Unit 3 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 15 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

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Check what stuck.

Units 1 through 3, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 42-question section, your choice. Even coverage means this is a retention check rather than a score estimate.

20 or 42questions

40topics

44codes covered

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