In the formula $\bar { x } \pm z _ { \mathfrak { w } / 2 } \sigma / \sqrt { n }$ , the $\alpha / 2$ refers to:
A) the probability that the confidence interval will contain the population mean.
B) the probability that the confidence interval will not contain the population mean.
C) the area in the lower tail or upper tail of the sampling distribution of the sample mean.
D) the level of confidence.

Question

Quizplus · Accepted Answer

The symbol $\alpha$ typically refers to the level of significance, while the formula given specifies $\alpha/2$ in terms of the tail areas in a standard normal distribution. Specifically, if we want to construct a confidence interval at level $1-\alpha$, we can find the z-score that corresponds to the upper $\alpha / 2$ tail area and the lower $\alpha / 2$ tail area (which add up to $\alpha$). These are denoted by $z_{\mathfrak{w}/2}$ in the formula, and they are used with the sample mean, standard deviation, and sample size to construct the confidence interval. So, the answer is C, which relates $\alpha/2$ to the tail area in the sampling distribution of the sample mean.

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In the Formula $\bar { x } \pm z _ { \mathfrak { w } / 2 } \sigma / \sqrt { n }$