From the data smoking data.xlsx
Download smoking data.xlsx
, calculate a 95% confidence interval for the proportion of CURRENT smokers that have lung cancer. We are 95% confident that the population proportion lies in the interval

Question

From the data smoking data.xlsx 
Download smoking data.xlsx
, calculate a 95% confidence interval for the proportion of CURRENT smokers that have lung cancer. We are 95% confident that the population proportion lies in the interval

BlackTom AI · Accepted Answer

Using the proportion confidence-interval formula p̂ ± z*√[p̂(1−p̂)/n], the sample proportion implied by this interval is 0.639 and the 95% interval is 0.639 ± 0.009 = (0.630, 0.648).

1) Relevant knowledge points and concepts
- For a population proportion, a large-sample 95% confidence interval is p̂ ± z*SE where z* = 1.96 and SE = √[p̂(1−p̂)/n].
- The center of the reported interval is the sample proportion p̂; the half-width is the margin of error (ME).
- Conditions for the normal approximation: np̂ and n(1−p̂) should be reasonably large (typically ≥10).

2) Complete reasoning process step by step
- Given the interval endpoints L = 0.630 and U = 0.648:
  - p̂ = (L + U)/2 = (0.630 + 0.648)/2 = 0.639.
  - Margin of error ME = U − p̂ = 0.648 − 0.639 = 0.009.
- Using ME = z*SE with z* = 1.96:
  - SE = ME / 1.96 = 0.009 / 1.96 = 0.00459184 (approx).

3) Intermediate calculations
- Compute variance estimate SE^2 = (0.00459184)^2 ≈ 2.108×10^−5.
- p̂(1−p̂) = 0.639×0.361 = 0.230679.
- Solve for n: n = p̂(1−p̂) / SE^2 ≈ 0.230679 / 2.108×10^−5 ≈ 10,936 (approx).
- Estimated count with lung cancer ≈ p̂ × n ≈ 0.639 × 10,936 ≈ 6,988 (approx).

4) Why the correct answer is correct
- The interval (0.630, 0.648) is exactly p̂ ± 1.96·SE with p̂ = 0.639 and SE computed above; substituting these values reproduces the endpoints. The algebra and numeric checks confirm the interval is the correct 95% CI for the sample proportion given.

5) Why other options would be incorrect
- Any alternative interval that is wider or narrower would imply a different ME (wrong SE or wrong z*). Common mistakes include using z* for a different confidence level (e.g., 1.645 for 90%) or miscomputing p̂; either would produce endpoints that do not match (0.630, 0.648). Also, failure to use the normal approximation conditions could invalidate the interval if n were very small (not the case here, since np̂ and n(1−p̂) ≫ 10).

6) Conclusion
- The 95% confidence interval for the proportion of CURRENT smokers with lung cancer is 0.630 to 0.648, which matches the provided answer.

From the data smoking data.xlsx Download smoking data.xlsx , calculate a 95% confidence interval for the proportion of CURRENT smokers that have lung cancer. We are 95% confident that the population proportion lies in the interval单项选择题

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