Inference from Sample Statistics and Margin of Error Flashcards

A random sample of 400 adults in a city finds that 64% have visited a museum in the past year, with a margin of error of ±4.7 percentage points at 95% confidence. Which of the following is the best interpretation of the margin of error?

Answer: We are 95% confident that the true population proportion is within 4.7 percentage points of 64%.

The margin of error defines the range around the sample proportion within which the true population proportion likely falls at the given confidence level.

A study reports a 95% confidence interval of ($48,000, $56,000) for the mean annual salary of teachers in a district. A union representative claims the mean salary is below $47,000. Which conclusion is most appropriate?

Answer: The claim is not supported because $47,000 is below the entire confidence interval.

Since $47,000 lies below the entire 95% confidence interval ($48,000–$56,000), the data do not support the representative's claim.

A researcher wants to estimate the proportion of adults who exercise daily. She collects three random samples of sizes 100, 400, and 1,600. Which sample will produce the widest 95% confidence interval?

Answer: The sample of 100

Smaller sample sizes produce larger margins of error and therefore wider confidence intervals; the sample of 100 will have the widest interval.

A survey of 900 randomly selected residents estimates that 41% support a new recycling program, with a margin of error of ±3.3 percentage points. A city official claims that support is at least 45%. Is this consistent with the survey?

Answer: No, because 45% is above the entire confidence interval (37.7%–44.3%).

The 95% CI is (41% − 3.3%, 41% + 3.3%) = (37.7%, 44.3%); since 45% lies above this interval, the official's claim is not supported.

A random sample of 100 customers gives a 95% confidence interval for mean purchase amount of ($42, $58). A store manager claims the true mean purchase is $60. Which assessment is most appropriate?

Answer: The claim is inconsistent with the data because $60 is outside the confidence interval.

Because $60 lies above the entire 95% confidence interval ($42–$58), the data do not support the manager's claim.

A survey company reports that their margin of error was reduced from ±4% to ±2% between two surveys. Which of the following best explains how this was achieved, assuming the confidence level remained the same?

Answer: The sample size was quadrupled.

Halving the margin of error (from 4% to 2%) requires quadrupling the sample size, since margin of error is proportional to 1/√n.

A 95% confidence interval for the proportion of adults who have completed a college degree is (0.30, 0.40). An education researcher claims the true proportion is 0.35. Which statement is most accurate?

Answer: The claim is consistent with the data because 0.35 falls within the confidence interval.

Since 0.35 lies within the 95% confidence interval (0.30–0.40), it is a plausible value for the true population proportion.