Inference from Sample Statistics and Margin of Error Flashcards

The table below shows results from two surveys about a proposed tax increase: Survey X: 600 respondents, 54% in favor, margin of error ±4% Survey Y: 150 respondents, 58% in favor, margin of error ±8% Which survey provides a more precise estimate of the proportion in favor, and why?

Answer: Survey X, because a larger sample produces a smaller margin of error.

Survey X uses a larger sample (600 vs. 150), yielding a smaller margin of error (±4% vs. ±8%), making it more precise.

A study estimates that the mean annual income of workers in an industry is $52,000, with a 95% confidence interval of ($49,500, $54,500). A labor economist claims the true mean income in that industry is $55,000. Which evaluation is most appropriate?

Answer: The claim is not supported because $55,000 falls above the upper bound of the interval.

Since $55,000 is above the upper bound of $54,500, it falls outside the confidence interval and is not a plausible value for the true mean.

A polling agency surveys 1,600 likely voters and reports that 49% plan to vote for a candidate, with a margin of error of ±2.5 percentage points. Which conclusion is best supported?

Answer: A majority support could exist because the confidence interval extends above 50%.

The confidence interval is 46.5%–51.5%; since this range includes values above 50%, majority support is a plausible population outcome.

A sample of 400 college students finds that 30% own a car. If the margin of error is ±4%, what is the 95% confidence interval, and which value is consistent with the population proportion?

Answer: 26%–34%; a true proportion of 31% is plausible.

The confidence interval is 30% ± 4% = 26%–34%; 31% falls within this range and is therefore a plausible population proportion.

To estimate the average price of a gallon of milk in a large city, an economist samples 100 stores and finds a mean of $3.85 with a margin of error of ±$0.20 at 95% confidence. If she wants the margin of error to be ±$0.10, approximately how many stores must she sample?

Answer: 400 stores

To halve the margin of error, you must quadruple the sample size (since MoE ∝ 1/√n): 100 × 4 = 400 stores.

A researcher reports that the 90% confidence interval for mean customer wait time is (4.2 min, 5.8 min). The 95% confidence interval using the same data would be:

Answer: Wider than (4.2 min, 5.8 min)

Increasing the confidence level requires a wider interval to ensure the parameter is captured more often under repeated sampling.

A random sample of 500 adults estimates that the mean number of books read per year is 12.4, with a standard error of 0.5. What is the approximate 95% confidence interval, using a critical value of 2?

Answer: (11.4, 13.4)

Margin of error = 2 × 0.5 = 1.0; the 95% confidence interval is 12.4 ± 1.0 = (11.4, 13.4).