Inference from Sample Statistics and Margin of Error Flashcards

A researcher wants to estimate the mean weight of a species of fish in a lake. Sample 1 consists of 50 randomly caught fish and sample 2 consists of 200 randomly caught fish. Both samples have similar means. Which sample will produce a confidence interval with a smaller width?

Answer: Sample 2, because larger samples produce narrower confidence intervals.

Larger samples reduce the standard error, resulting in narrower confidence intervals and more precise estimates.

The results of a study report: 'We are 95% confident that the mean number of hours adults spend reading per week is between 3.2 and 4.8 hours.' What is the margin of error?

Answer: 0.8 hours

The margin of error is half the confidence interval width: (4.8 − 3.2)/2 = 0.8 hours.

A random sample of 64 students shows a mean study time of 3.5 hours per day with a standard deviation of 1.2 hours. If the sample size is increased to 256 students with the same standard deviation, by what factor does the margin of error change?

Answer: It is divided by 2.

Sample size increases by a factor of 4 (64 to 256), so √4 = 2; the margin of error is divided by 2.

A television network surveys 1,000 randomly selected viewers and reports that 63% watch a particular show, with a margin of error of ±3 percentage points at 95% confidence. Which of the following populations can this estimate validly be generalized to?

Answer: All viewers in the population from which the random sample was drawn

Inferences from a random sample can only be generalized to the population from which that sample was drawn, not to broader populations not represented.

A poll estimates that 48% of voters support a candidate, with a margin of error of ±4 percentage points. A competitor's poll estimates 51% support with a margin of error of ±4 percentage points. Which conclusion is most appropriate?

Answer: The results are consistent because the confidence intervals overlap.

Poll 1 gives 44%–52% and Poll 2 gives 47%–55%; these intervals overlap, so both results are consistent with a range of true values.

A company surveys 400 customers and finds a mean satisfaction score of 7.4 out of 10, with a margin of error of ±0.3. A manager claims the true mean satisfaction score is at most 7.0. Is this claim consistent with the survey?

Answer: No, because 7.0 is below the lower bound of the confidence interval.

The confidence interval is 7.1 to 7.7; since 7.0 is below the lower bound of 7.1, it is not a plausible value for the true mean.

A study uses a random sample to estimate the proportion of adults in a county who exercise regularly. The 95% confidence interval is (0.34, 0.46). Which of the following must be true?

Answer: The sample proportion is 0.40.

The sample proportion is the midpoint of the confidence interval: (0.34 + 0.46)/2 = 0.40.