Find the critical value, z α 2 z_{\frac{\alpha}{2}} , for a 80% confidence interval.

Here, we're asked to find the critical value, z alpha over 2, for an 80% confidence interval. So the first thing we want to do here is find alpha over 2. So this is going to be 1 minus c, our confidence level, over 2. Now in this case, since we have this confidence level of 80%, this is 1 minus 0.8 over 2, which ultimately gives us an alpha over 2 value of 0.1. Now remember that from this point, there are 2 ways that you could go about this. You could look this value up in a z table or you could use your calculator. Now if you are using your calculator, remember that this is the area underneath one of our tails here. So if you're looking for this z alpha over 2 value, the positive one on the other side here, you also need to include all of this area here. So if this area is 0.1, that's what we just found, and this area here is our confidence level, 0.8, that means when you type this into your calculator, you need to note that your area here is 0 point 9. That's these two values added together because, remember, this is the total area to the left of your boundary. Now if you are not using a calculator, you can just look up this 0.1 in the body of your z table, and it will give you the negative version of your critical value. Now either way that you choose to do this, you should end up getting an answer here that z alpha over 2 is equal to 1.282 and this is our final answer. Feel free to let us know if you have any questions, and let's keep going.

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

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Multiple Choice

Find the critical value, $z sub alpha over 2$ , for a 80% confidence interval.

0.10

1.282

0.40

0.90

Verified step by step guidance

Understand that the critical value, z_{\frac{\alpha}{2}}, is used in constructing confidence intervals for a population mean when the population standard deviation is known.

Recognize that for a confidence level of 80%, the significance level $ \alpha $ is 0.20, because $ \alpha = 1 - \text{confidence level} $.

Calculate $ \frac{\alpha}{2} $ to find the tail probability for the critical value. For $ \alpha = 0.20 $, $ \frac{\alpha}{2} = 0.10 $.

Use the standard normal distribution table or a calculator to find the z-score that corresponds to the cumulative probability of 0.90 (since 1 - 0.10 = 0.90). This z-score is the critical value, z_{\frac{\alpha}{2}}.

Verify the critical value by checking the z-score table or using statistical software to ensure it matches the expected value for a 80% confidence interval.