Significance Test for a Proportion — FRQ Walkthrough
A free-response question in exam style, fully worked — then the scoring logic behind every point.
The question · original, exam-style
A city bike-share program claims that at least 65% of its trips end at a different station than they started at. A transportation blogger suspects the true percentage is lower. She pulls a random sample of 240 completed trips from the program's public data and finds that 144 of them ended at a different station. Is there convincing statistical evidence, at the α = 0.05 level, that fewer than 65% of all trips end at a different station? Complete the appropriate inference procedure, showing your reasoning.
Written by Marlio as an analog of the exam format. Not a College Board question.
Model response
State: Let p = the true proportion of all bike-share trips that end at a different station. H₀: p = 0.65 versus Hₐ: p < 0.65.
Plan: One-sample z-test for a proportion. Random — the 240 trips are stated to be a random sample. 10% — 240 trips is less than 10% of all trips the program has recorded. Large counts, using p₀: n·p₀ = 240(0.65) = 156 ≥ 10 and n(1 − p₀) = 240(0.35) = 84 ≥ 10. All conditions are met.
Do: p̂ = 144/240 = 0.60. z = (0.60 − 0.65) / √(0.65·0.35/240) = −0.05 / √0.000948 = −0.05 / 0.0308 ≈ −1.62. The p-value is P(Z < −1.62) ≈ 0.0526.
Conclude: Because the p-value 0.0526 is greater than α = 0.05, we fail to reject H₀. There is not convincing evidence that fewer than 65% of all bike-share trips end at a different station.
How it’s scored — point by point
- POINT 1
States the hypotheses and defines the parameter.
The grader must see exactly what p refers to and which way the alternative points. Naming p as a population proportion (not the sample) and writing Hₐ: p < 0.65 earns this; writing hypotheses about p̂, or a two-sided alternative, loses it.
- POINT 2
Names the correct procedure and verifies all conditions.
You must identify the one-sample z-test for a proportion and check random, 10%, and large counts — showing the actual numbers 156 and 84, not just naming the conditions. A condition listed without evidence earns no credit.
- POINT 3
Correct mechanics: test statistic and p-value.
The standard error has to use p₀ = 0.65, giving z ≈ −1.62 and a left-tail p-value ≈ 0.053. Building the standard error from p̂, or reporting a two-tailed p-value, forfeits this point even when the arithmetic is otherwise clean.
- POINT 4
Conclusion in context, linked to the p-value and α.
A complete conclusion compares the p-value with α, states the decision (fail to reject H₀), and answers the original question about bike-share trips. A bare "reject / fail to reject" with no context and no comparison to α is incomplete.
Common point-losers
- Writing the hypotheses about p̂ (the sample) instead of p (the population).
- Using p̂ = 0.60 in the standard error instead of the null value p₀ = 0.65.
- Naming the conditions but not backing them with the actual counts.
- Saying "we accept H₀" — you can only fail to reject it.
- Giving a decision with no context, e.g. "reject H₀" with no mention of bike-share trips.
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