In Exercises 27–30, find the critical values and for the level...

Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.98, n = 25

Accepted Answer

Welcome back, everyone. Let's take a look at this next problem. For a confidence level of C equals 0.99. And a sample size of N equals 18, find the critical values, the right-sided chi squared critical value, and the left sided chi squared critical value. So, looking up chi squared values, first, let's start with the graph, which is always helpful to keep things in mind. And recall that with the chi squared distribution, we do not have a symmetrical distribution. So, when we look for these critical values, we can't, we're not going to have, you know, a plus or minus like we do with, for instance, critical Z values. So remember that we look at a chi square chart, it's always giving us the area to the right of the critical value. So let's draw on here, just a theoretical. Left sided chi squared. And right sided chi squared critical value. So lines on the left and right side, and we're looking for those outside. Regions under the curve on the left side and the right side. So we're looking at a two-sided test. And we want to start with our alpha value. So, our alpha value will be 1 minus the confidence value, or 1 minus 0.99. Which is equal to 0.01. But since we have a two-sided test, on the right side, that area underneath the curve, that area on the right side is going to be alpha divided by 2, since we have two tails. So that area will equal 0.01 divided by 2, so it's going to equal 0.005. And now, although my distribution looks a little more symmetrical than it should, it isn't the greatest um chi square distribution, but we know the theory here. Then when we look at how our table works on a chi square table. We're going to look for the critical value on the left side that corresponds to Where the area to the right of that critical value. So I'm, I'm right showing drawing blue lines covering that area. So that includes our right tail, everything to the right, because again, we want to sort of delineate that region on the left. And now that probability will be equal to. Because the area of that rejection region on the left, just. It's also going to be alpha divided by 2. So that means that the area under the curve to the right, so I'll draw a little box with lines through it, so that striped area. is going to be 1 minus alpha divided by 2. Which will equal 0.995. It's getting a little crowded there. So that makes sky square. We always have to think. Carefully through when we're looking for a left-sided critical chi square value. But drawing that graph is always helpful to keep in mind what values you're looking for. So, now we're ready to go to our chi square table. So, what are the numbers we need? We're going to need our alpha. On the table, I'll put in parentheses table, which will be the area we're looking for that corresponds with our critical value. And we're going to need our degrees of freedom, and that equals N minus 1. We're told we have a sample size of 18, so that's going to equal 17. So in the case of our right sided. Critical chi squared value. That area. Is equal to 0.05. And again, our degrees of freedom are 17, and when we refer to the table, we're going to get a right-sided critical value of 35.718. And then on the left side, left sided critical chi squared value. The area, again, we're looking to that area to the right, so that's going to be 0.995. And our degrees of freedom, equal to 17. So again, use area as the alpha on the table. And that will give us a critical value of 5.697. And that makes sense. You have this low chi squared value for your left value, and you have a high chi squared value for your right. Always good to go do a gut check and make sure you didn't make a mistake somewhere in your math. But once again, with our confidence level of 0.99, sample size of N equals 18, we need to find the right and left-sided chi squared critical value, and we have our right side value of 35.718, and our left-sided critical value of 5.697. See you in the next video.

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.
c = 0.98, n = 25

Key Concepts

Critical Values

Level of Confidence

Sample Size

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