In the context of constructing a confidence interval for the mean of a normal distribution with known variance, which function is used to evaluate the probability that a sample mean falls within a certain range of the population mean?

• A. The cumulative distribution function $F\left(x\right)$ of the standard normal distribution
• B. The moment generating function $M\left(t\right)$ of the exponential distribution
• C. The probability mass function $p\left(x\right)$ of the binomial distribution
• D. The cumulative hazard function $H\left(x\right)$ of the Weibull distribution

Answer: (A)

Verified step by step guidance

1. Understand that when constructing a confidence interval for the mean of a normal distribution with known variance, we are interested in the probability that the sample mean falls within a certain range around the population mean.
2. Recall that the sample mean of a normal distribution with known variance is itself normally distributed, and we often standardize it to a standard normal variable using the formula: \(Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}}\), where \(\overline{X}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
3. To find the probability that the sample mean falls within a certain range, we need to evaluate the probability \(P(a \leq \overline{X} \leq b)\), which translates to \(P\left( \frac{a - \mu}{\sigma / \sqrt{n}} \leq Z \leq \frac{b - \mu}{\sigma / \sqrt{n}} \right)\) after standardization.
4. This probability is found using the cumulative distribution function (CDF) of the standard normal distribution, denoted as \(F(z)\), which gives the probability that the standard normal variable is less than or equal to \(z\).
5. Therefore, the function used to evaluate the probability that the sample mean falls within a certain range of the population mean is the cumulative distribution function \(F(x)\) of the standard normal distribution.