Inference from Sample Statistics and Margin of Error Flashcards

A researcher finds that a 95% confidence interval for mean exam scores is (72, 88). A school board member says, 'There's a 95% chance the true mean is in this interval.' Is this statement correct?

Answer: No, because once an interval is calculated, the true mean is either in it or not — it's the method that has 95% success rate.

The 95% refers to the procedure: 95% of all confidence intervals constructed this way contain the true mean; once a specific interval is computed, it either contains the true mean or it does not.

A random sample of 400 light bulbs from a manufacturer has a mean lifetime of 1,200 hours with a standard error of 20 hours. The manufacturer claims bulbs last more than 1,250 hours on average. What does the 95% confidence interval (using critical value ≈ 2) suggest about this claim?

Answer: The claim is not supported because 1,250 is above the confidence interval (1,160–1,240).

The 95% CI is 1,200 ± 40 = (1,160, 1,240); since 1,250 is above the upper bound, it is not a plausible population mean, so the manufacturer's claim is not supported.

A study of 400 randomly selected households finds a mean monthly water usage of 3,400 gallons with a margin of error of ±150 gallons. The city's target average usage is 3,600 gallons. Is the target consistent with the sample data?

Answer: No, because 3,600 gallons falls above the upper bound of the confidence interval.

The confidence interval is 3,400 ± 150 = (3,250, 3,550); since 3,600 is above the upper bound, it is not plausible given the sample.

A survey reports: 'We are 95% confident that between 34% and 42% of adults exercise daily.' What is the sample proportion and margin of error from this report?

Answer: Sample proportion = 38%, margin of error = 4%

Sample proportion = midpoint = (34 + 42)/2 = 38%; margin of error = half-width = (42 − 34)/2 = 4%.

Two different analysts use the same random sample of 500 observations to compute confidence intervals, but Analyst A uses 90% confidence while Analyst B uses 99% confidence. Which statement is true?

Answer: Analyst B's interval is wider because higher confidence requires a larger critical value.

Higher confidence levels use larger critical values (e.g., z* = 2.576 for 99% vs. 1.645 for 90%), resulting in a wider interval to capture the true parameter more reliably.

A poll of 625 randomly selected residents shows 48% oppose a new development. The margin of error is ±4%. A city council member says this proves that fewer than half of all residents oppose the development. Is this conclusion valid?

Answer: No, because the confidence interval (44%–52%) includes values above 50%, making majority opposition plausible.

The confidence interval of 44%–52% includes values above 50%, so majority opposition (>50%) is a plausible population outcome; the council member's conclusion is not supported.

A study estimates from a random sample of 100 runners that the mean marathon completion time is 4 hours 12 minutes, with a margin of error of ±18 minutes. Another researcher claims the mean time is under 3 hours 50 minutes. Is this claim plausible?

Answer: No, because 3 hours 50 minutes (230 min) is well below the lower bound of the confidence interval.

The confidence interval is 252 ± 18 = (234, 270) minutes; 230 minutes (3:50) is below the lower bound of 234, making the claim implausible.