Confidence Interval 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 regional library wants to estimate the proportion of its cardholders who used the e-book service at least once in the past month. Staff take a random sample of 180 cardholders and find that 63 of them used the e-book service in that window. Construct and interpret a 95% confidence interval for the true proportion of all cardholders who used the e-book service at least once in the past month.
Written by Marlio as an analog of the exam format. Not a College Board question.
Model response
Identify: One-sample z-interval for a proportion. p̂ = 63/180 = 0.35.
Conditions: Random — the 180 cardholders are stated to be a random sample. 10% — 180 is less than 10% of all cardholders. Large counts — n·p̂ = 63 ≥ 10 and n(1 − p̂) = 117 ≥ 10. Conditions met.
Calculate: For 95% confidence, z* = 1.96. The standard error is √(0.35·0.65/180) = √0.001264 ≈ 0.0356, so the margin of error is 1.96·0.0356 ≈ 0.070. The interval is 0.35 ± 0.070 = (0.280, 0.420).
Interpret: We are 95% confident that the true proportion of all library cardholders who used the e-book service at least once in the past month is between 0.280 and 0.420.
How it’s scored — point by point
- POINT 1
Identifies the procedure and checks the conditions.
Name the one-sample z-interval for a proportion and verify random, 10%, and large counts with the actual counts 63 and 117. This is the point most often lost, by skipping the large-counts check.
- POINT 2
Correct mechanics: critical value, standard error, and interval.
Uses z* = 1.96 for 95%, builds the standard error from p̂, and reports the interval (0.280, 0.420). A wrong critical value (for example 1.645) or a standard error built from a wrong p̂ loses this point.
- POINT 3
Interprets the interval in context.
The interpretation must be about the true population proportion of cardholders, use the phrase "95% confident," and give the endpoints in context. Interpreting it as a probability about one interval, or describing 95% of cardholders, earns no credit.
Common point-losers
- Interpreting the interval as "there's a 95% probability the true proportion is in (0.280, 0.420)" — the interval is fixed once computed.
- Saying "95% of cardholders" instead of describing the plausible values for the single true proportion.
- Skipping the large-counts condition, or listing it without the counts.
- Using the critical value for the wrong confidence level.
- Forgetting to define the proportion in context — of what population, over what time window.
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