ETF1100 Chap.7 Inference for Proportions
Inference for Proportions
Inference for Proportions focuses on binary business outcomes and the sample share p-hat. It separates interval and test standard errors, checks expected success and failure counts, selects the test direction from the claim, and reports effects in percentage points. Denominator choice, repeat observations and selection bias remain essential even when the arithmetic is short.
What this chapter covers
- 01
Binary outcomes and success definitions
- 02
Sample and population proportions
- 03
Proportion confidence intervals
- 04
Null standard errors
- 05
One-proportion z tests
- 06
Percentage-point effects and conditions
Worked example · free
Test a conversion proportion
- 1State H0: p = 0.40 and H1: p greater than 0.40.
- 1Compute p-hat = 94 / 200 = 0.470.
- 1Compute null SE = sqrt(0.40 x 0.60 / 200) = 0.03464.
- 1Compute z = (0.47 - 0.40) / 0.03464 = 2.02.
- 1Use p about 0.0217 and reject at 5%.
- 1Report evidence above 40% and a 7 percentage-point observed lift.
Key terms
- Population Proportion
- The share of a population satisfying a clearly defined binary outcome.
- Sample Proportion
- The observed success count divided by the sample size.
- Binary Outcome
- A variable with two defined results, coded as success and failure for analysis.
- Null Standard Error
- The proportion-test standard error calculated from the benchmark proportion under the null.
- Percentage Point
- The direct subtraction of two percentages.
Inference for Proportions FAQ
Why do tests and intervals use different proportion inputs?
A test asks what sample proportions would look like if the null benchmark were true, so its standard error uses p0. An interval estimates around the observed proportion, so its usual standard error uses p-hat. Mixing them answers the wrong sampling question.
How do I choose the denominator for a rate?
Define success and identify every unit eligible to experience that outcome. Conversion may use eligible visits, defect rate inspected units and complaint rate completed orders. Excluding inconvenient non-successes inflates the proportion and changes the target population.
What is the difference between percentage points and percent change?
Subtracting 40% from 47% gives 7 percentage points. Relative percent change divides that gap by the 40% baseline, giving 17.5%. State the scale explicitly because the two numbers answer different business questions.
How do repeated visits affect a proportion analysis?
If one person contributes several visits, rows may not provide independent information and a visit-level proportion is not automatically a customer-level proportion. State the observational unit, investigate repeat identifiers and align the denominator with the intended claim. Ignoring clustering can make uncertainty appear smaller than the design supports.
Exam move
Define success in words before writing the fraction and mark the eligible denominator. Practise converting the same result between a proportion, percentage and rate per fixed exposure, following the entry convention exactly. Keep a visible split between interval and test calculations: observed p-hat for the usual interval standard error, null p0 for the one-sample test standard error.
Before using a normal approximation, check expected successes and failures with the appropriate proportion. Translate the claim into a left, right or two-sided alternative before viewing the sample. Report the observed difference in percentage points and compare it with operational cost rather than stopping at significance.
When observations can repeat for the same customer or come from an opt-in sample, state the design limitation. On exam questions, re-read whether the requested number is a decimal or percentage and round only after the final calculation. Construct a four-column setup for every problem: eligible units, counted outcome, sample proportion and comparison benchmark.
Reconcile success plus failure with the total before running any method. Write interval and test standard errors on separate lines and annotate why one uses the observed share while the other assumes the null value. For a test, compute expected successes and failures under the benchmark; for an interval, inspect the observed counts.
If either side is sparse, flag that the normal approximation may not be reliable rather than forcing the formula. Practise changing only the claim wording among greater than, less than and different from so the appropriate tail becomes automatic. After the decision, convert the effect into percentage points, relative change and expected event count at a realistic workload, then choose the scale most useful to management.
Distinguish a visit-level proportion from a customer-level proportion whenever one person can appear repeatedly. Examine whether exclusions, opt-in behaviour or missing outcomes alter the denominator. For a rate per thousand, keep the exposure unit visible and do not call it a percentage.
In timed work, estimate whether the observed share lies above or below the benchmark, preserve full precision and check that the submitted format matches the prompt. Finish with the represented population, effect magnitude, evidence strength and a specific limitation.
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