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

Here we're asked to find the critical value z alpha over two for an 80% confidence interval. So the first thing we wanna do here is find alpha over two. So this is gonna be one minus c, our confidence level, over two. Now in this case, since we have this confidence level of 80%, this is one minus 0.8 over two, which ultimately gives us an alpha over two value of 0.1. Now remember that from this point, there are two ways that you could go about this. You could look this value up in 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 two 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 0.9. 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 two 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.

Find the critical value, zα2z_{\frac{\alpha}{2}}z2α, for an 80% c...

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

Statistics for Business

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

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Multiple Choice

Find the critical value, $z_{\frac{\alpha}{2}}$ , for an 80% confidence interval.

0.10

1.282

0.40

0.90

Verified step by step guidance

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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Find the critical value, zα2z_{\frac{\alpha}{2}}z2α, for an 80% confidence interval.

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Find the critical value, $z_{\frac{\alpha}{2}}$ , for an 80% confidence interval.