Inference from Sample Statistics and Margin of Error Flashcards

A random sample of 500 households in a county found that 38% have solar panels installed, with a margin of error of ±4 percentage points at 95% confidence. Which of the following intervals represents the plausible range for the true proportion of all county households with solar panels?

Answer: 34% to 42%

The confidence interval is formed by adding and subtracting the margin of error from the sample proportion: 38% ± 4% gives 34% to 42%.

A research team surveys 900 randomly selected adults and finds that 61% support a proposed policy. The team then conducts a second survey using a random sample of 225 adults. Assuming the true proportion stays the same, how does the margin of error for the second survey compare to the first?

Answer: It doubles

Margin of error is proportional to 1/√n; reducing n from 900 to 225 (dividing by 4) doubles 1/√n, so the margin of error doubles.

A 95% confidence interval for the mean hours of weekly exercise among adults in a city is (3.1 hours, 4.5 hours). A fitness company claims the true mean is 5 hours per week. Which statement best evaluates this claim?

Answer: The claim is inconsistent with the data because 5 hours is outside the confidence interval.

Because 5 hours falls outside the 95% confidence interval of (3.1, 4.5), the data do not support the company's claim.

A school principal wants to estimate the proportion of students who walk to school. She surveys a random sample of 50 students and obtains a margin of error of ±7%. If she wants to reduce the margin of error to ±3.5%, approximately how large must the new sample be?

Answer: 200

Halving the margin of error requires quadrupling the sample size; 50 × 4 = 200.

In a study of 800 randomly chosen shoppers, 44% said they prefer buying groceries online. The reported margin of error is ±3.5 percentage points at 95% confidence. A store manager claims that fewer than 40% of shoppers prefer online grocery shopping. Which statement best assesses this claim?

Answer: The claim is plausible because 40% falls just inside the lower boundary of the confidence interval.

The 95% CI is approximately 40.5% to 47.5%, so 40% barely falls outside it, making the manager's claim not well-supported by the data.

A 95% confidence interval for a population proportion is reported as (0.52, 0.68). What is the sample proportion and the margin of error?

Answer: Sample proportion = 0.60; margin of error = 0.08

The sample proportion is the midpoint: (0.52 + 0.68)/2 = 0.60, and the margin of error is half the width: (0.68 − 0.52)/2 = 0.08.

A researcher surveys a random sample of 1,024 adults and finds that 30% report feeling stressed daily, with a margin of error of ±3 percentage points at 95% confidence. If the researcher increases the sample to 4,096 adults, what would the new margin of error be, assuming the same proportion?

Answer: ±1.5 percentage points

Quadrupling the sample size halves the margin of error: 3% ÷ 2 = 1.5 percentage points.