Confidence Intervals and Estimation Flashcards

A polling company reports that 54% of voters favor a candidate with a margin of error of ±3%. This means the true proportion likely falls between:

Answer: 51% and 57%

The confidence interval spans from 54% − 3% = 51% to 54% + 3% = 57%.

For a confidence interval for a proportion, what value of p̂ produces the widest interval for a given sample size and confidence level?

Answer: p̂ = 0.5

The expression p̂(1−p̂) is maximized when p̂ = 0.5, producing the largest standard error and widest interval.

A researcher wants a margin of error of no more than 2 points with 95% confidence and estimates σ = 10. What minimum sample size is needed?

Answer: 196

n = (z·σ/E)² = (1.96 × 10 / 2)² = 9.8² ≈ 96.04, rounded up to 97; closest standard answer is 97, but using exact formula gives 97 — select 97.

Which critical value corresponds to a 90% two-sided confidence interval using the z-distribution?

Answer: 1.645

A 90% interval leaves 5% in each tail, and z₀.₀₅ = 1.645.

A confidence interval is described as 'capturing' the true parameter. This means:

Answer: The interval contains the true population value.

An interval 'captures' the parameter when the true value falls somewhere between its lower and upper bounds.

If a 95% confidence interval for μ is (50, 70), can we conclude that μ = 45?

Answer: No, because 45 is outside the interval and is inconsistent with the data at this confidence level.

Values outside the confidence interval are implausible at the given confidence level based on the sample data.

An estimator is said to be unbiased if:

Answer: Its expected value equals the population parameter.

Unbiasedness means that on average across all samples, the estimator equals the true parameter value.