Take a box containing 100 0's and 10 1's. We take 40 draws with replacement and calculate the proportion of 1's to be 0.1.

What is the approximate 68% Confidence Interval for the population proportion of 1's (to 2dp)?

Question

Take a box containing 100 0's and 10 1's. We take 40 draws with replacement and calculate the proportion of 1's to be 0.1.

What is the approximate 68% Confidence Interval for the population proportion of 1's (to 2dp)?

BlackTom AI · Accepted Answer

For a sample proportion, the 68% (≈1σ) confidence interval is p̂ ± 1·SE, where SE = sqrt(p̂(1 − p̂)/n).

1. Relevant knowledge points and concepts
- The sampling distribution of a sample proportion p̂ is approximately normal for moderate n, with mean p and standard error SE = sqrt(p(1 − p)/n).  
- A 68% confidence interval corresponds approximately to ±1 standard error (z ≈ 1).  
- Sampling with replacement yields independent Bernoulli trials so no finite-population correction is needed.

2. Complete reasoning process step by step
- Observed sample proportion p̂ = 0.1, sample size n = 40.  
- Compute the standard error using the sample estimate: SE = sqrt(p̂(1 − p̂)/n).  
- The 68% CI = p̂ ± 1·SE.

3. Calculations with intermediate steps
- p̂(1 − p̂) = 0.1 × 0.9 = 0.09.  
- Divide by n: 0.09 / 40 = 0.00225.  
- SE = sqrt(0.00225) = 0.0474341649 (≈ 0.04743).  
- Lower bound = 0.1 − 0.047434 = 0.052566 ≈ 0.0526.  
- Upper bound = 0.1 + 0.047434 = 0.147434 ≈ 0.1474.  
- Rounding both to two decimal places: lower → 0.05, upper → 0.15.

4. Why the correct answer is correct
Using the normal approximation and the 1-SE rule for a 68% interval yields (0.0526, 0.1474), which rounds to (0.05, 0.15). Sampling with replacement justifies using the binomial-based SE formula. The arithmetic above directly produces the stated interval.

5. Why other options are incorrect
There are no alternate options provided here; any other interval that does not equal p̂ ± SE (rounded) would contradict the standard 68% normal-approximation interval computed from the observed data.

6. Conclusion
The approximate 68% confidence interval for the population proportion is (0.05, 0.15), which matches the provided answer.

Take a box containing 100 0's and 10 1's. We take 40 draws with replacement and calculate the proportion of 1's to be 0.1. What is the approximate 68% Confidence Interval for the population proportion of 1's (to 2dp)?单项选择题

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