Question

the national assessment of educational progress (naep) includes a mathematical test for eighth - grade students. scores on the test range from 0 to 500. suppose that you give the naep test to an srs of 2500 eighth - graders from a large population in which the scores have mean μ = 282 and standard deviation σ = 110. the mean (\bar{x}) will vary if you take repeated samples. suppose that we computed a 90% confidence interval for μ. which is true? (for this question, explain in detail why your choice is correct and each of the other alternatives is not in q29.)
a. this 90% confidence interval would have a smaller margin of error than the 95% confidence interval
b. this 90% confidence interval would have a larger margin of error than the 95% confidence interval
c. this 90% confidence interval could have either a smaller or a larger margin of error than the 95% confidence interval. this varies from sample to sample.
d. this 90% confidence interval would have the margin of error as the 95% confidence interval.

Explanation:

Step1: Recall Confidence Interval Margin of Error

The margin of error for a confidence interval (when population standard deviation \(\sigma\) is known) is given by \( E = z^* \times \frac{\sigma}{\sqrt{n}} \), where \( z^* \) is the critical value, \(\sigma\) is the population standard deviation, and \( n \) is the sample size.

Step2: Analyze Critical Values for Confidence Levels

For a 90% confidence interval, the \( z^* \)-value (critical value) is smaller than the \( z^* \)-value for a 95% confidence interval. For example, \( z^* \approx 1.645 \) for 90% confidence and \( z^* \approx 1.96 \) for 95% confidence.

Step3: Compare Margins of Error

Since the margin of error \( E \) is directly proportional to \( z^* \) (when \(\sigma\) and \( n \) are constant), a smaller \( z^* \) (for 90% confidence) will lead to a smaller margin of error compared to the 95% confidence interval. The sample size \( n = 2500 \) and \(\sigma = 110\) are constant here, so the only change is in \( z^* \). Thus, the 90% confidence interval has a smaller margin of error than the 95% confidence interval.

Answer:

a. This 90% confidence interval would have a smaller margin of error than the 95% confidence interval