Lesson 13: Confidence Intervals for a Population ProportionAP

A sample proportion is a good guess at p, but a guess with a known margin of error is far better. A confidence interval packages the estimate together with its precision: estimate ± margin of error. CED Unit 6 introduces the one-proportion z-interval, and with it the four-step framework — State, Plan, Do, Conclude — that organises every inference response on the AP exam from here on.

Anatomy of the interval

For a population proportion: p̂ ± z* × √(p̂(1−p̂)/n). The multiplier z* is the critical value from the standard normal distribution for the chosen confidence level (90% → 1.645, 95% → 1.960, 99% → 2.576). The term √(p̂(1−p̂)/n) is the standard error of p̂ — note carefully that it uses p̂, not p: in the real world we do not know p (that is the point of the interval), so we estimate the spread with the sample value. This is the one deliberate difference from Lesson 11's sampling-distribution model, and examiners probe it.

Conditions (Plan step — all three must be checked, every time):

• Random: the data come from a random sample (or randomised experiment) — inference from biased samples is invalid no matter how pretty the arithmetic.
• Normal: the sampling distribution of p̂ is approximately normal, checked with observed counts: np̂ ≥ 10 and n(1−p̂) ≥ 10.
• Independent: observations are independent; when sampling without replacement, verify the 10% condition (n ≤ 10% of the population).

Worked example 1. A random survey of 400 voters finds 236 supporting a ballot measure, so p̂ = 236/400 = 0.59. Standard error = √(0.59 × 0.41/400) ≈ 0.0246. For 95% confidence: 0.59 ± 1.96 × 0.0246 = 0.59 ± 0.048, giving (0.542, 0.638). Conditions: random sample (stated), successes 236 ≥ 10 and failures 164 ≥ 10, and 400 voters is surely under 10% of the electorate. Conclusion in context: "We are 95% confident that the interval from 0.542 to 0.638 captures the true proportion of all voters who support the measure." That sentence — naming the parameter, in context, with the confidence level attached to the *procedure* — is the template for every interval conclusion on the exam.

What "95% confident" actually means

The confidence level describes the method, not this particular interval: if we repeated the sampling and interval construction many times, about 95% of the intervals produced would capture the true p. The true p is fixed; it is the interval that jumps around from sample to sample. So "there is a 95% probability that p is in (0.542, 0.638)" is wrong under the AP framework — p is either in there or it is not; the 95% is our confidence in the process. Readers deduct for the probability phrasing every year.

Controlling the margin of error

The margin of error shrinks when: n grows (like 1/√n — to halve the ME, quadruple the sample), the confidence level drops, or p̂ is far from 0.5. To plan a study for a target ME at the design stage, solve n ≥ p*(1−p*) × (z*/ME)² using a guessed p* — or p* = 0.5 if no guess is available, since p(1−p) is maximised at 0.5 (the conservative choice).

Worked example 2. A campaign wants a 95% interval with margin of error no more than 0.03 and no prior estimate of support. With p* = 0.5: n ≥ 0.25 × (1.96/0.03)² = 0.25 × 4,268.4 ≈ 1,067.1, so survey at least 1,068 voters. This is why real polls settle near n ≈ 1,000: it is the sample size that buys ±3 points at 95% confidence. Note the rounding direction: always round n up — rounding down would miss the target precision.

Common mistakes

• The probability interpretation. Never write "95% probability the true proportion is in this interval" — write "we are 95% confident..." and think process.
• Skipping or mumbling conditions. "Conditions are met" earns nothing; name each condition and show the check (numbers for Large Counts, a sentence for Random and 10%).
• Using p in the standard error (there is no p available) or the sampling-distribution SD from Lesson 11.
• A conclusion with no context — "p is between 0.542 and 0.638" without saying *the proportion of all voters who support the measure* loses the conclusion point.
• Believing a wider interval is "worse data." A 99% interval is wider because it demands more confidence; width trades against confidence, not accuracy of the sample.

AP Exam Experience

Question 1. A 95% confidence interval for the proportion of students who ride the bus is (0.31, 0.43). Which interpretation is correct?

• (A) 95% of students have proportions between 0.31 and 0.43.
• (B) There is a 0.95 probability that the true proportion of all students who ride the bus is between 0.31 and 0.43.
• (C) We are 95% confident the interval (0.31, 0.43) captures the true proportion of all students who ride the bus.
• (D) 95% of samples of this size give exactly this interval.
• (E) The true proportion is 0.37 with 95% confidence.

Answer: (C). Confidence attaches to the method's long-run capture rate; the phrasing "we are 95% confident the interval captures the true [parameter in context]" is the only acceptable form. (B) is the classic error — the parameter is fixed, so no probability attaches to it being in this computed interval.

Question 2. A researcher computes a 90% confidence interval for p from a sample. If she recomputes at 99% confidence using the same data, the new interval will be:

• (A) Narrower, because higher confidence is more precise
• (B) Wider, because a larger critical value is used
• (C) The same width, because the data are unchanged
• (D) Centred at a different value
• (E) Invalid, because confidence level cannot be changed after sampling

Answer: (B). z* grows from 1.645 to 2.576, so the margin of error — and the interval — widens. More confidence costs more width; the centre p̂ is unchanged.

Question 3. A random sample of 120 students contains 18 who are left-handed. Which check correctly verifies the Normal condition for a one-proportion z-interval?

• (A) np = 120 × 0.5 ≥ 10
• (B) Successes = 18 ≥ 10 and failures = 102 ≥ 10
• (C) n = 120 ≥ 30
• (D) 120 is less than 10% of all students
• (E) The sample is random

Answer: (B). The Normal (Large Counts) condition for intervals uses *observed* counts: np̂ = 18 successes and n(1−p̂) = 102 failures, both at least 10. Choice (D) is the Independence/10% condition, (E) is the Random condition — real conditions, but not the one asked about.

Free-response question. A school district wants to estimate the proportion of its students who walk or cycle to school. A random sample of 85 students finds 62 who walk or cycle. (a) Verify the conditions for constructing a one-proportion z-interval. (b) Construct a 90% confidence interval for the true proportion. (c) Interpret the interval in context. (d) A board member says: "This means 90% of students walk or cycle." Explain the error.

Scoring guideline.

• Part (a): Essentially correct if all three are addressed: Random (stated in the problem — say so); Normal: np̂ = 62 ≥ 10 and n(1−p̂) = 23 ≥ 10 ✓; Independent: 85 students is less than 10% of the district's students (accept a reasonable statement that the district has more than 850 students). Partially correct if one condition is omitted or the Large Counts check uses p instead of counts.
• Part (b): Essentially correct if p̂ = 62/85 ≈ 0.729, SE = √(0.729 × 0.271/85) ≈ 0.0482, z* = 1.645 for 90%, interval 0.729 ± 0.079 ≈ (0.650, 0.809). Partially correct if the method is right with one arithmetic slip. Incorrect if z* = 1.96 is used (that is the 95% value — a favourite trap when the problem says 90%).
• Part (c): Essentially correct for "We are 90% confident that the interval from about 0.650 to 0.809 captures the true proportion of all students in the district who walk or cycle to school" — confidence language, interval, and parameter in context all present. Partially correct if "probability" replaces "confident" or context is missing.
• Part (d): Essentially correct if it explains that 90% is the confidence level (the proportion of intervals constructed this way that would capture the truth), not a percent of students; the interval itself estimates the percent of students at roughly 65% to 81%. Incorrect if it merely restates the interval without addressing the confusion.

Deeper dive: the margin of error in the wild

When a news report says "±3 percentage points", it is quoting (usually) the 95% margin of error for p̂ near 0.5 with n ≈ 1,068 — the exact computation of Worked Example 2. Three subtleties journalists routinely drop: (1) the quoted ME applies to the full sample; a subgroup (say, 120 young voters) has a much larger ME (about ±9 points), so "candidate surges among the young" stories are usually noise; (2) the ME quantifies *only* random sampling error — bias from bad frames, nonresponse and wording (Lesson 9) sits entirely outside it, like a miscalibrated scale reporting weight to the nearest gram; (3) when two candidates' percentages are compared, the ME of the *difference* is larger than each individual ME — a 2-point lead inside a ±3 poll is properly described as a statistical tie. Carry these three caveats and you can dismantle most election-season headlines in a sentence each — which is, not coincidentally, what the exam's data-analysis FRQs keep asking students to do with survey results.

Second worked example: buying more confidence

Return to the voter survey of Worked Example 1 (p̂ = 0.59, n = 400, SE ≈ 0.0246) and rebuild the interval at 99% confidence: the margin of error becomes 2.576 × 0.0246 ≈ 0.063, giving (0.527, 0.653) — noticeably wider than the 95% interval (0.542, 0.638). Suppose a campaign strategist asks: "At 99% confidence, can we still say a majority supports the measure?" Yes: the entire 99% interval lies above 0.50, so the data support the majority claim at that stricter standard too. But had p̂ been 0.54, the 95% interval (0.491, 0.589) would straddle 0.50 and the majority claim would fail at 95% — a fragility the single number p̂ conceals. This is the practical grammar of intervals: the confidence level is the price of the claim, the width is what you pay, and statements about thresholds (majorities, legal limits, claimed values) are settled by whether the threshold falls inside or outside — the duality with tests, visible here in its natural habitat.