Inference from Sample Statistics and Margin of Error Flashcards

A survey of 300 randomly selected adults found that 72% exercise at least once per week, with a margin of error of ±5 percentage points at 95% confidence. Which of the following statements is a valid interpretation of this result?

Answer: We are 95% confident that between 67% and 77% of all adults exercise at least once per week.

A 95% confidence interval means we are 95% confident the true population proportion falls in the interval formed by the sample proportion ± margin of error.

A random sample of 256 workers finds a mean weekly salary of $720 with a standard deviation of $80. A different sample of 64 workers from the same population finds a mean of $715 with the same standard deviation. Which sample produces a smaller margin of error, and why?

Answer: The sample of 256, because larger samples produce smaller margins of error.

With a larger sample size and the same standard deviation, the standard error (and thus the margin of error) is smaller for the sample of 256.

A study estimates the proportion of students who own a car at a large university. A 95% confidence interval based on a sample of 400 students is (0.28, 0.36). What is the margin of error and the sample proportion?

Answer: Margin of error = 0.04; sample proportion = 0.32

The sample proportion is the midpoint (0.28 + 0.36)/2 = 0.32 and the margin of error is half the width (0.36 − 0.28)/2 = 0.04.

A city survey reports that 55% of residents support a new transit plan, with a margin of error of ±6 percentage points. A local politician says this proves majority support. Which response best evaluates this claim?

Answer: The claim is somewhat supported but uncertain; the interval (49%–61%) includes values below 50%.

The confidence interval of 49%–61% includes values below 50%, so majority support is plausible but not certain.

A researcher is comparing two methods of estimating average household income. Method A uses a random sample of 100 households; Method B uses a random sample of 400 households. Both are from the same population. Which method produces a more reliable estimate, and by what factor is its margin of error smaller?

Answer: Method B; its margin of error is half that of Method A.

Method B uses 4 times the sample size, so its margin of error is 1/√4 = 1/2 that of Method A.

A 95% confidence interval for the mean test score of students in a district is (74, 82). A state official claims the district mean is 85. Which of the following most accurately describes this situation?

Answer: The official's claim is inconsistent with the sample data.

Because 85 lies above the entire 95% confidence interval (74, 82), the official's claim is not supported by the sample data.

A researcher reports a 95% confidence interval of (0.41, 0.59) for the proportion of voters who prefer Candidate X. Which statement is best supported by this interval?

Answer: The data do not provide convincing evidence that Candidate X has majority support.

The interval (0.41, 0.59) crosses 0.50, meaning values both above and below 50% are plausible, so no definitive conclusion about majority support can be drawn.