Yun wants to build a one-sample \( z \) interval with \( 82 \% \) confidence to estimate what proportion of users will click an advertisement that appears on his website. He takes a random sample of 200 users and finds that 34 of them clicked the advertisement. What critical value \( z^{*} \) should Yun use to construct this confidence interval? Choose 1 answer: (A) \( z^{*}=0.842 \) (B) \( z^{*}=0.915 \) (C) \( z^{*}=1.282 \) (D) \( z^{*}=1.341 \) ( 1.31

To find the critical value \( z^{*} \) for an 82% confidence interval, we need to determine the z-scores that correspond to the tails of the normal distribution. Since the confidence level is 82%, the area in the tails combined is 1 - 0.82 = 0.18. Thus, each tail will have an area of 0.09. Using the standard normal distribution table or calculator, we look for the z-score that corresponds to an area of 0.09 in the left tail, which leaves an area of 0.91 in the right tail. The corresponding z-score for 0.91 is approximately 1.341. So, the critical value \( z^{*} \) that Yun should use is \( 1.341 \). Now let's check for the potential common mistakes. One such mistake when calculating a critical value is forgetting to account for both tails when determining the area for the confidence interval. It’s essential to split the remaining area evenly between the two tails to identify the proper z-score correctly. It's also easy to confuse the area covered by the confidence level with total area, leading to erroneous conclusions!