L-INF-4: Solutions — Confidence Intervals for a Proportion

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Count, Then Decide (C1 + C5)

Each generated instance supplies a random sample size , a population size , and the percentage who answered yes. Use the values displayed in your instance.

Steps 1 and 2 — the observed outcome counts

Step 3 — the decision

The stem states simple random sampling and , so randomness and approximate independence pass. The decision rests on the counts: build the interval only when both and . About one instance in three fails: either fewer than 10 successes or fewer than 10 failures. Then this lesson’s interval is not adequately licensed; collect more data or use a method built for sparse counts, such as the Wilson interval of Challenge 1.

Problem 2 — Standard Error and Interval Construction (C2 + C3 + C4)

For the displayed , , and confidence level:

Step 1 — point estimate

Step 2 — estimated standard error

Step 3 — critical value

The activity asks for as its own typed step; it is never printed. Using the course-rounded critical values:

Step 4 — margin of error

Step 5 — endpoints

The activity accepts the interval rounded to four decimals. Compute the endpoints from the unrounded and standard error, then round the final bounds.

Problem 3 — Which Proportion Goes In? (C2 + C7)

• Case A builds an interval from data already collected, so it uses the observed :

• Case B plans a survey with no earlier estimate, so it uses the planning value , which gives the largest and never undershoots:

Neither case uses the true : it is the unknown being estimated. In L-INF-6 a third value appears, the hypothesized .

Problem 4 — Practical Interpretation of Confidence (C6)

The reported interval is .

Correct interpretation: Using this method, we are 95% confident that the true proportion of adults who support stricter food-labelling laws lies between 57.7% and 66.3%.

The distractors fail for different reasons:

• “95% probability that is in this interval” assigns probability to a fixed parameter after the interval has been computed.
• “95% of the 500 adults” describes sample members rather than the population parameter.
• “95 of 100 future sample proportions” describes future statistics landing in one fixed range, not the long-run capture rate of the interval-producing method.

Problem 5 — Determining Required Sample Size (C7)

Every generated instance supplies a confidence level, target margin , and population size . About half the instances supply a prior estimate; the others say no earlier estimate exists. The result is a minimum under that planning value, the course-rounded critical value, and the large-population approximation; if the eventual sample proportion is closer to 0.5, the realized margin can be larger.

Use the critical value that matches the displayed confidence level. When a prior estimate is given, use it as rather than replacing it with 0.50; when none exists, use . After rounding up, verify before calling the result the required sample size under this course method.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Build a Confidence Interval (C1 + C2 + C4 + C5)

The named sample units, target population, outcome, , , , and confidence level change each time; neither nor is a round number, and the sampling fraction varies. The sample units align with the target population, and the invariant calculation is:

The generator states random sampling, constrains and both observed outcome counts to clear 10, and reports the endpoints to four decimals. Its contextual interpretation is:

Using this method, we are [confidence level]% confident that the true proportion of [named target population] who or that [named outcome] lies between the displayed lower and upper bounds.

Problem 2 — Margin of Error and Interpretation (C3 + C6)

For the displayed instance:

Enter rounded to four decimals. If you also form the interval:

The generated survey states a simple random sample, population size, sample units, target population, and measured outcome. Interpret the endpoints for that named population proportion. Interpret the confidence level as the method’s approximate long-run capture rate under comparable sampling conditions. Do not say that fixed has that probability of lying in this one completed interval.

Problem 3 — Can This Survey Support the Interval? (C5)

Seven of twelve cards are shown each time. Sort by two questions in order.

1. Was the sample random? Volunteers answering an online poll, call-in listeners, a researcher’s own classes and the first shoppers through a door are not random samples. No interval method repairs that, however many people answered.
2. Are there at least 10 of each outcome? 6 defective bottles out of 150, 9 reactions out of 2,000 patients, 9 students without a bike out of 40 and 9 blemished apples out of 300 all leave one side below 10: collect more data or use a method for sparse counts. A random sample with at least 10 of each (38 bike-share users out of 1,200; 32 of 64; 47 of 220; 14 of 80) gets this lesson’s interval.

Problem 4 — Determine the Required Sample Size (C7)

First determine the planning value:

Then compute:

About half the instances call for . An estimate from a different city five years ago is not defensible, so it is treated as no estimate. The activity grades the ceiling of the full-precision .

Problem 5 — The Price of Precision (C7)

With no estimate, and . The confidence level is drawn from 90%, 95% and 99%, and the first target from ±4, ±5 and ±6 points; the next two rows halve it twice. For example, at 90% with targets ±4, ±2 and ±1 points:

Each halving of multiplies by 4. Every minimum rounds up.

Problem 6 — Can the Headline Say “Majority”? (C6)

Compare 0.50 with the printed 95% interval:

• 0.50 below the whole interval: the poll supports a majority headline;
• 0.50 inside the interval: majority support is not established (values on both sides of one half are plausible);
• 0.50 above the whole interval: support is below half.

The last blank is always the same: the ±E covers only random sampling error. Non-response, people who could not be reached, and question wording can add error the margin does not count.

Problem 7 — Diagnose a Survey Report (C2 + C3 + C5 + C6 + C7)

Each report has five steps and exactly one error, drawn from a proportion-specific taxonomy; the error can sit in any step:

1. Conditions: using instead of checking that both outcome counts reach 10.
2. Standard error: using the planning value 0.5 in place of the observed .
3. Margin and interval: reporting the full width as the margin of error; the margin is half of it.
4. Interpretation: assigning a probability to the fixed proportion after the interval is computed.
5. Follow-up plan: rounding the required sample size down instead of up.

An error that changes a number is carried forward consistently, so the first wrong step is the only error.

Problem 8 — Multi-Step Survey Decision (C1 + C2 + C3 + C4 + C5 + C6 + C7)

The sample has and from a population of 12,000.

(a) Conditions

The stem states simple random sampling.

All conditions are satisfied.

(b) Two-sided 95% interval

The 95% interval is approximately .

(c) Interpretation and claim

Using this method, we are 95% confident that the true proportion of adults who meet the guideline lies between about 33.8% and 42.3%. The entire interval is below 45%, so the data support the “fewer than 45%” claim at the corresponding confidence level. They do not prove it with certainty.

(d) Future sample size

Mixed Review A — Sampling Distribution of a Mean (Generated Family)

The generated stem now explicitly states a normally distributed population and an independent random sample. Therefore:

For the displayed threshold :

The generator constructs from one of its listed values and grades the resulting right-tail probability to four decimals.

Mixed Review B — Sample Size for a Mean (Generated Family)

For the displayed known , confidence level, and target margin :

Square the entire ratio and round up. This is the L-INF-2 mean-planning formula, not the proportion formula used in this lesson.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Question 1 — Feynman Test

The theoretical standard error depends on unknown :

Because is unknown, we replace it with the observed :

When sampling is random, observations are reasonably independent, and both observed outcome counts are at least 10, the standard z procedure is usually a workable approximation. If those conditions fail, use another method or collect more data.

“95% confident” describes the interval-producing method: across many comparable random samples, it aims to capture fixed about 95% of the time. It is not a 95% probability statement about after this one interval has been computed.

Question 2 — May the Interval Be Built?

The generated survey either passes or fails the count condition. Check randomness and (both stated), then the two observed counts and : build the interval only when both reach 10. A wrong option that cites or a t-interval names a rule for means; for a yes/no answer fixes both centre and spread, so no separate standard deviation is estimated.

Question 3 — Cold Interval

The stem does not name the method. A random sample, small relative to , with at least 10 of each answer licenses the one-proportion z-interval. Then:

with chosen from the stated level (1.645, 1.96 or 2.576). The steps are available after your solo attempt.

Question 4 — Plan the Survey

As in Independent Problem 4: use a defensible estimate when one is given, otherwise ; compute and round up.

Question 5 — Fact-Check the Headline

As in Independent Problem 6: the headline is supported only when 0.50 is below the whole interval, not established when 0.50 is inside it, and wrong when the whole interval is below 0.50. The option claiming the margin covers every source of error is always wrong.

Extra Practice — Student-Services Planning

A student-services office in a hypothetical 30,000-student multi-campus system selects a simple random sample of 640 students. Of those sampled, 256 prefer later evening service hours. The office asks for a defensible 99% range for the population proportion and wants to plan a future study with margin of error at most 0.025. Identify the parameter and estimate, verify the conditions, calculate and interpret the 99% range, and use the current sample proportion to find the minimum future sample size.

The response is binary and the target is a population proportion, so use a two-sided z-interval for .

(a) Parameter and estimate

is the true proportion of all 30,000 enrolled students in the multi-campus system who prefer later service hours.

(b) Conditions

The stem states simple random sampling.

All conditions are satisfied.

(c) Two-sided 99% interval

Using the course-rounded critical value:

Using this method, we are 99% confident that between about 35.0% and 45.0% of all enrolled students prefer later service hours.

(d) Future sample size

The planned sample remains within the independence guideline:

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Both paths draw fresh numbers on every rematch, so these solutions give the method with one worked instance each.

📰 The Newsroom

Task 1 — the range. A random poll of a few thousand adults from a province of millions is a tiny share of the population, and both answers have far more than 10 people, so the one-proportion z-interval applies. For example, with 571 of 1,068 in favour:

The 95% interval is approximately .

Task 2 — the note to the editor.

• Headline: 0.50 below the whole interval supports “majority”; 0.50 inside the interval does not establish it; a whole interval below 0.50 means support is below half.
• The 18–24 story: check the subgroup’s own counts. With 31 of 38 in favour, only 7 are opposed, fewer than 10, so this lesson’s ± interval should not be reported for the subgroup. With 61 of 96 in favour (35 opposed) the interval is licensed, but its margin comes from : , about ±9.6 points. A subgroup never borrows the full poll’s margin.
• The ±: it covers only random sampling error.

📋 The Pollster

Task 1 — the target. Nobody has polled the question, so . For ±2 points at 95%:

Compare with the budget: a budget of 2,000 interviews is 401 short.

Task 2 — what the budget buys. With fixed by the budget and no estimate:

For 2,000 interviews at 95%, , about ±2.2 points.

Task 3 — the reply. The “we only poll Quebec, so we need fewer people” idea is wrong: contains no population size. Quebec, Canada and the United States all need the same for the same margin; the population’s size matters only when the sample is a large share of it (beyond the 10% check), and then the adjustment needs fewer interviews, not more. The advice follows the budget: raise it to the required , or go ahead if it already covers it.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Challenge 1 — The Wilson Score Interval

The generator draws a small or extreme sample with from 8 to 25 and equal to 1, 2, , or . The prompt supplies the two-sided 95% level, , and the Wilson formulas below. For the displayed values:

The standard two-sided 95% z-interval is:

For the Wilson interval, first compute:

Then:

The generated family deliberately violates this lesson’s success–failure condition. Its comparison illustrates that the standard interval can leave the parameter space , whereas the Wilson score interval remains bounded.

Challenge 2 — The Shape of the Margin of Error

For and 95% confidence:

(a) Values

(b) Maximum

is maximized at because is maximized there. A Bernoulli outcome has its greatest variance when success and failure are equally likely.

(c) Planning claim

Using is not conservative for every .

For every true proportion strictly between 0.3 and 0.7, the variance factor is greater than 0.21, so planning with 0.7 understates the required sample size there, including for values below 0.5 such as 0.4. With no defensible prior information, use .

Challenge 3 — Two Polls, One Question

Each generated instance supplies two independent simple random samples from the same named population of size during the same period, a common confidence level, and counts and . Before constructing the intervals, verify for each firm that , , and ; the generator licenses all three checks.

Firm A

Firm B

Informal overlap check

The intervals overlap when:

If the intervals overlap, judging by eye is inconclusive: two 95% intervals can overlap even when the two proportions differ significantly (Schenker and Gentleman, 2001), so overlap does not show the polls agree. If they do not overlap, the difference is larger than sampling luck usually produces, which is strong evidence (not proof) that the polls measured different proportions. Settling the question formally needs a two-sample procedure for , beyond this lesson.