Bluebook SAT Test Inference from Sample Statistics and Margin of Error 7

Question 1

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?

Accepted 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.

Answer

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.

Question 2

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?

Accepted Answer

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

Answer

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.

Question 3

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?

Accepted Answer

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

Answer

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.

Question 4

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?

Accepted Answer

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

Answer

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

Question 5

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?

Accepted Answer

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

Answer

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.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Inference from Sample Statistics and Margin of Error 7