Question

which of the following is not a requirement for constructing a confidence interval for estimating a population mean with σ known?
choose the correct answer below.
○ a. the sample is a simple random sample.
○ b. either the population is normally distributed or n > 30, or both.
○ c. the confidence level is 95%.
○ d. the sample measures a quantitative value.

Explanation:

• Option A: A simple random sample is a fundamental requirement for constructing a confidence interval. It ensures that each member of the population has an equal chance of being selected, which is crucial for the validity of statistical inferences.
• Option B: When the population standard deviation \(\sigma\) is known, if the population is normally distributed, we can construct a confidence interval. If the population is not normally distributed, by the Central Limit Theorem, for \(n>30\) (a large - sample size), the sampling distribution of the sample mean \(\bar{x}\) is approximately normal. So, either the population is normally distributed or \(n > 30\), or both is a requirement.
• Option C: The confidence level does not have to be \(95\%\). We can construct confidence intervals for different confidence levels such as \(90\%\), \(99\%\) etc. The formula for the confidence interval for the population mean \(\mu\) when \(\sigma\) is known is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\), where \(z_{\alpha/2}\) changes depending on the confidence level.
• Option D: Since we are estimating a population mean (a quantitative measure), the sample must measure a quantitative value. If the sample measured a non - quantitative (qualitative) value, we could not calculate a meaningful mean for the purpose of this confidence interval.

Answer:

C. The confidence level is 95%.