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 = 0.90, n = 16

Accepted Answer

Hello everyone. Glad to have you with us. Let's take a look at this next problem. Our problem says, for a confidence level C equals 0.80, and a sample size N equals 20, find the critical values of chi squared sub L and chi squad subR. So the right and left-sided critical values. And our answer choices are A, the left is 28.412, the right is 11.651. B, the right is 27.204, the left is 11.591. To see, the right is 27.204, the left is 11.651. Or the left is 28.412 and the right is 11.591. So, let's think about what we're looking for here. We're talking about a two-tailed test, since we have both a left and right-sided critical value. So, for chi squared, sketch a sample, chi squared distribution. We have our chi squad sub L, chi squad subR being those two values on the left and on the right. Where we're going to be looking for that region. That falls to the outs and under the curve to the outside of those two values, that will be our rejection region. So, I've highlighted that in blue. And I'm given a confidence level, so from that, I can obtain an alpha value, which is 1 minus C. So that will equal 1 minus 0.80, so our alpha is going to be 0.2. But that means that the total area under both tails. Is 0.2. So the area under one tail is alpha divided by 2. So we always have to remember that. Now, one thing, if I were on a test, so I might be rushing, maybe I wouldn't have time to complete some problems. We know that our right-sided value must be greater than our left sided value. So we can actually eliminate some of these multiple choice options on that alone. So in choice A, the left sided value is greater than the right sided value. So that's incorrect. So put greater on top of the left side about you. So choice B, that's all right. Choice C, that's all right. But choice D again shows a left-sided value larger than the right sided value. So, Cho D, we can also roll out. So just a little hint, how on a test you'd be able to get down to two answer choices like that if you were in a rush. Now, a right-sided value in the case of chi squared is a bit simpler. So we'll want to think about what would we look up on the table for that chi squared sub R. Well, our alpha on the table will be equal to alpha divided by 2, cause that's the area of one tail. So that's going to be 0.2 divided by 2, so 0.1. So we'll be looking for the chi squared value where alpha is 0.1. And then our Degrees of freedom, we know, degrees of freedom are N minus 1. Our sample size is 20, so our degrees of freedom are 19. So, the two eyes we look up in the table are for alpha, 0.1, and degrees of freedom, 19. And when we look that up, that's going to give us a result of 27. 204. Well, both B and C have that value, so we need to go ahead and make sure that we Look at that chi squared, the left-sided value. Now, the left is always a bit trickier because the way the chi squared table is organized, it always looks at a right-sided arrangement. So, the chi squared value you get is reflecting the area to the right side of that critical value. So, I'm going to draw lines, so it's this entire part of the curve to the right. Of our left squared critical value, including the right tail. So the area. Of that striped Area Well, we know that the little tail that's outside of that area is alpha divided by 2. So, the area of the stripes is 1 minus alpha divided by 2. That'll be 1 minus 0.1, and that's going to equal 0.9. So, to look up our left-sided critical value, we'd want to look up in the chi square table. The value where alpha is 0.9. And then 19 degrees of freedom. And that's going to give us 11.651. And that makes sense. We have our right side of value greater than our left side of value. And that's going to correspond. 2 C. Chi squared sub L is 11.651. Choice B has the wrong value for a left-sided critical value. So, once again, we're given a confidence level and a sample size, and we want to know our left and right sided critical values of chi squared. And we've worked that through. And calculated that our answer is choice C. Our chi squared on the right is 27.204, or the critical value on the right. Our left-sided critical value for chi squared is 11.651. 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 = 0.90, n = 16

Key Concepts

Critical Values

Level of Confidence

Sample Size

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