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

Yes, because confidence intervals are probability statements about the true parameter.

Answer

Yes, but only if the sample was random.

Answer

No, because the true mean equals the sample mean in large samples.

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 claim is supported because 1,250 is within 2 standard errors.

Answer

The claim is supported because the sample mean is 1,200.

Answer

The claim is inconclusive without knowing the full population data.

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

No, because 3,600 is more than the sample mean.

Answer

Yes, because 3,600 gallons falls within the confidence interval of 3,250–3,550 gallons.

Answer

Yes, because confidence intervals are never used to assess targets.

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 = 34%, margin of error = 8%

Answer

Sample proportion = 42%, margin of error = 8%

Answer

Sample proportion = 38%, margin of error = 8%

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

Analyst A's interval is wider because lower confidence captures more values.

Answer

Both intervals have the same width because the sample is identical.

Answer

Analyst A's interval is more likely to contain the true parameter.

Question 6

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?

Accepted Answer

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

Answer

Yes, because 48% is less than 50%.

Answer

Yes, because the sample proportion is the best single estimate.

Answer

No, because polls can only be used to show support, not opposition.

Question 7

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?

Accepted Answer

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

Answer

Yes, because 3 hours 50 minutes is close to the sample mean.

Answer

Yes, because the sample only included 100 runners.

Answer

No, because marathon times are always above 4 hours.