Suppose we sample 100 people, and 72 of them support a particular candidate for political office. We would like to use this data to estimate
𝑝
, the proportion of all people in the population who support the candidate.

What is the estimated standard error of
𝑃
^
?

Question

Suppose we sample 100 people, and 72 of them support a particular candidate for political office.  We would like to use this data to estimate 
𝑝
, the proportion of all people in the population who support the candidate.

What is the estimated standard error of 
𝑃
^
?

BlackTom AI · Accepted Answer

The standard error of a sample proportion p̂ is sqrt[p̂(1 − p̂)/n], so we estimate it using the sample proportion in place of the unknown true p.

1. Relevant knowledge points and concepts
- For a sample of size n from a large population, the sampling distribution of the sample proportion p̂ has standard error SE(p̂) = sqrt[p(1 − p)/n]. When p is unknown we use p̂ in the formula: SE ≈ sqrt[p̂(1 − p̂)/n]. This comes from the binomial variance np(1 − p) divided by n^2.

2. Complete reasoning process, step by step
- Compute the sample proportion: p̂ = number supporting / n = 72/100.
- Plug p̂ and n into the SE formula and evaluate.

3. Calculations with intermediate steps
- p̂ = 72/100 = 0.72.
- Compute p̂(1 − p̂) = 0.72 × (1 − 0.72) = 0.72 × 0.28 = 0.2016.
- Divide by n: 0.2016 / 100 = 0.002016.
- Take square root: SE = sqrt(0.002016) ≈ 0.0449 (rounded to four decimal places).

4. Why the correct answer is correct
- The computed SE = 0.0449 directly follows from the standard formula for the standard error of a proportion using the observed p̂ = 0.72 and sample size n = 100. This number measures the typical sampling variability of p̂ around the true population proportion.

5. Why the other options are incorrect
- 0.0020: This is approximately the quantity 0.2016/100 = 0.002016 (the variance), not its square root; it is the variance of p̂, not the standard error.
- 0.2016: This is p̂(1 − p̂) before dividing by n; it is not the SE.
- 0.4490: This is ten times the correct SE (decimal moved one place); it does not follow from the formula.

6. Conclusion
- The estimated standard error of p̂ is 0.0449, which matches the provided correct answer.

Suppose we sample 100 people, and 72 of them support a particular candidate for political office. We would like to use this data to estimate 𝑝 , the proportion of all people in the population who support the candidate. What is the estimated standard error of 𝑃 ^ ?单项选择题

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