Question
Elena wants to build a one-sample \( z \) interval to estimate what proportion of computers produced at a factory have a certain defect. She chooses a confidence level of \( 94 \% \). A random sample of 200 computers shows that 12 computers have the defect. What critical value \( z^{*} \) should Elena use to construct this confidence interval? Choose 1 answer: (A) \( z^{*}=1.55 \) (B) \( z^{*}=1.75 \) (C) \( z^{*}=1.88 \) (D) \( z^{*}=1.96 \)
Ask by Dickson Bradley. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Elena should use \( z^{*} = 1.88 \) to construct the confidence interval.
Solution
To determine the appropriate critical value for a 94% confidence level, we first note that the total area in both tails is 6% (since 100% - 94% = 6%). Since the distribution is symmetric, each tail has 3% (0.03) of the area. We need the z-value such that 97% (1 - 0.03) of the distribution is to the left of it.
Looking up the z-value for a cumulative probability of 0.97 (or using a calculator), we find that approximately:
z* ≈ 1.88
Thus, the correct answer is (C) z* = 1.88.
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Simplify this solution The Deep Dive
To find the critical value \( z^{*} \) for a \( 94 \% \) confidence level, we’ll look for the value that leaves \( 3 \% \) in each tail of the standard normal distribution (since \( 100\% - 94\% = 6\% \), and half of that is \( 3\% \)). Using a standard \( z \)-table or a calculator, we find that \( z^{*} \approx 1.88 \) corresponds to a tail probability of \( 3\% \). Thus, the answer is: (C) \( z^{*}=1.88 \)
