Finding Critical Values for χ2 In Exercises 3–8, find the crit...

Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.90, n = 8

Accepted Answer

Hello everyone. Glad to have you back. Here's the next problem. Find the critical values, and these are the chi squared critical value on the right, and the chi squared critical value on the left for a confidence level of C equals 0.95, and a sample size of N equals 12. So, let's think about what we need here. We will need our degrees of freedom for a chi square test. So, if N is 12, we call that degrees of freedom, DF equals 1, excuse me, not 1, N minus 1, getting that backwards. So, 12 minus 1 is 11, so we have 11 degrees of freedom. And then we'll need our alpha value. We're given a confidence level. So we recall that alpha. equals 1 minus 0.95. So, our alpha is equal to 0.05. Now, the one tricky thing here is that, of course, usually we do, uh, when we do chi squared distributions, we look only at the right tailed value, but we're asked for the critical value um of the left tailed. Side of the chi square distribution. So crossword on the right is simple enough. We have a two-tailed situation here. So we have to remember that. The area under the tail on either side. I'll say the area. In each tail is equal to alpha divided by 2. So that's going to be 0.025. So to find our chi squared value. On the right, the right tailed chi square value. This is quite straightforward. So, we look on our chi square table using 0.025 as the alpha, for the purposes of the table, and 11 degrees of freedom. And when we look that up on the table, give you a second if you want to pause and look at it for yourself. We get 21.920. The chi squared value on the left just requires a little bit of an extra step. We recall that because the chi square distribution is not symmetrical like the normal distribution, we can't just look forward on the other side. We need to calculate for the table. We know that we were looking at this 0.025. And because the chi squared table is a right tailed table, that would mean that the value it gives us the critical value, everything to the right of that represents. Or excuse me, represents a probability of less than. Alpha, or in this case, alpha divided by 2. So then to look for our left critical our critical value on the left side, then we would have to subtract 1 minus 0.025. Which will give us 0.975. Which in fact gives us the probability of a value being on the right side of that critical value, then we know that There's a 0.025. Probability of being on the left side of that critical value. So that's why that's our left-sided critical value, and why we need to look this up separately when we're doing chi squared. So we use our cri square table and use 0.975 as the alpha, and then we use those 11 degrees of freedom 11. We'll get our left-sided critical value as being 3.816. So, there we have it. We needed to find the left-sided and right-sided critical chi squared critical value. When we had a confidence level of 0.95, and a sample size of N equals 12, and we have our right-sided chi squared critical value of 21.920, and our left sided chi square critical value of 3.816. See you in the next video.

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.
c = 0.90, n = 8

Key Concepts

Chi-Square Distribution

Critical Values

Degrees of Freedom

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