Multiple Choice
Which of the following best describes the purpose of a confidence interval in statistics?
7. Sampling Distributions & Confidence Intervals: Mean
Introduction to Confidence Intervals
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Find the critical value, , for an 80% confidence interval.
A
0.10
B
1.282
C
0.40
D
0.90
Verified step by step guidance1
Identify the confidence level for the interval, which is 80%. This means that the level of significance, \( \alpha \), is 20% or 0.20.
Since the confidence interval is two-tailed, divide the level of significance by 2 to find \( \frac{\alpha}{2} \). So, \( \frac{\alpha}{2} = \frac{0.20}{2} = 0.10 \).
Determine the critical value \( z_{\frac{\alpha}{2}} \) using the standard normal distribution table or a calculator. This value corresponds to the point where the cumulative probability is \( 1 - \frac{\alpha}{2} = 0.90 \).
Look up the cumulative probability of 0.90 in the standard normal distribution table to find the \( z \)-score. This \( z \)-score is the critical value \( z_{\frac{\alpha}{2}} \).
Verify the critical value by checking that it matches the expected value for an 80% confidence interval, which is approximately 1.282.
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