Confidence Intervals and Estimation Flashcards

A 95% CI and a 99% CI are both computed from the same data. Which statement is true?

Answer: The 99% CI is wider.

Higher confidence requires a larger critical value, making the 99% interval necessarily wider than the 95% interval.

Which of the following scenarios involves point estimation?

Answer: Stating the average test score is 76.

Stating a single value (76) as the estimate of the population mean is an example of point estimation.

In a t-distribution with 15 degrees of freedom, what happens to the critical value compared to the z critical value for the same confidence level?

Answer: The t critical value is larger.

The t-distribution has heavier tails than the z-distribution, so its critical values are always larger for the same confidence level and finite degrees of freedom.

A student calculates a confidence interval of (30, 50) for a mean. The sample mean used was:

Answer: 40

The sample mean is the midpoint of the interval: (30 + 50) / 2 = 40.

A factory claims its bolts have a mean diameter of 10mm. A 95% CI from a sample is (9.5, 9.8). What can be concluded?

Answer: The factory's claim is inconsistent with the sample data at 95% confidence.

Since 10mm falls outside the 95% CI (9.5, 9.8), the sample data contradict the factory's claim at this confidence level.

What does the Central Limit Theorem guarantee for large samples that makes confidence intervals valid?

Answer: The sampling distribution of the mean will be approximately normal.

The CLT ensures that for large n, the distribution of sample means is approximately normal regardless of the population's shape.

A 95% confidence interval for a proportion is computed as (0.62, 0.78). The margin of error is:

Answer: 0.08

Margin of error = half the interval width = (0.78 − 0.62) / 2 = 0.16 / 2 = 0.08.