Question

when finding the margin of error for the mean of a normally distributed population from a sample, what is the critical probability, assuming a confidence level of 58%?
○ 0.21
○ 0.42
○ 0.58
○ 0.79

Explanation:

Step1: Recall confidence level and critical probability

The confidence level (CL) is the probability that the interval estimate contains the population parameter. The critical probability (\(p^*\)) for the margin of error in a confidence interval for the mean (when dealing with the normal distribution) is related to the confidence level. The formula to find \(p^*\) is \(p^* = 1-\frac{1 - \text{CL}}{2}\).

Step2: Substitute the confidence level

Given the confidence level (CL) is \(58\%=0.58\). First, calculate \(1 - \text{CL}=1 - 0.58 = 0.42\). Then, \(\frac{1 - \text{CL}}{2}=\frac{0.42}{2}=0.21\). Finally, \(p^*=1 - 0.21 = 0.79\)? Wait, no, wait. Wait, actually, the critical probability for the z-score (or t-score) in the margin of error is the probability that the test statistic is within the critical value. Wait, maybe I mixed up. Wait, the confidence level is the area in the middle. So the area in each tail is \(\frac{1 - \text{CL}}{2}\). So the critical probability (the cumulative probability up to the upper critical value) is \(\text{CL}+\frac{1 - \text{CL}}{2}=\frac{\text{CL}+ 1}{2}\)? Wait, no, let's think again. For a confidence interval, the total area in the two tails is \(1 - \text{CL}\). So the area in one tail is \(\frac{1 - \text{CL}}{2}\). The critical probability (the probability that the variable is less than or equal to the upper critical value) is \(1-\frac{1 - \text{CL}}{2}\). Let's plug in \(\text{CL} = 0.58\):

\(1-\frac{1 - 0.58}{2}=1-\frac{0.42}{2}=1 - 0.21 = 0.79\). Wait, but let's check the options. Wait, maybe I made a mistake. Wait, the critical probability here: Wait, the margin of error for the mean (when using z or t) involves the critical value, which corresponds to a probability. Wait, the confidence level is 58%, so the area between the two critical values is 0.58. So the area to the left of the upper critical value is \(0.58+\frac{1 - 0.58}{2}=0.58 + 0.21 = 0.79\). So the critical probability (the probability associated with the upper critical value) is 0.79. Wait, but let's verify with the formula. The formula for the critical probability (the probability that the standard normal variable is less than the critical z - value) is \(p^*=1-\frac{1 - \text{CL}}{2}\). So with \(\text{CL}=0.58\), \(p^* = 1-\frac{1 - 0.58}{2}=1 - 0.21 = 0.79\). So the critical probability is 0.79.

Answer:

0.79 (the option with 0.79)