In Problems 5–8, find the critical values χ²₁₋ᵅ⁄₂ and χ²ᵅ⁄₂ fo...

Question

In Problems 5–8, find the critical values χ²₁₋ᵅ⁄₂ and χ²ᵅ⁄₂ for the given level of confidence and sample size.

8. 99% confidence, n = 14

Accepted Answer

Hi, everybody. Let's move on to our next problem. Find the critical values chi squared. Sub 1 minus alpha divided by 2, and chi squared subalpha divided by 2, for a confidence level of 95%, with sample size N equals 15. And then we have our two Chris score values listed, um, in each of our our choices, in each of our choices A through D. Um, they're lifted in the order with, first the chi squared value for 1 minus alp over 2, and then the chi squared value for alp over 2. So, I'm just going to read the values rather than reading out the long chi squared description. So for choice A, it's 5.629 and 26.119. For B, 26.119, and 5.629. For C 27.587 and 8.62, and for D 29.141 and 9.390. So, to figure out what our alpha is going to be, we have our confidence level. So we recall that alpha is equal to 1 minus the confidence level. So, 1 minus 0.95. So alpha equal to 0.05. And that means the value of alpha divided by 2 is equal to 0.025. If our sample size is 15, the other thing we need are degrees of freedom, which equal N minus 1, so that equals 14. And then let's recall the meaning of these chi squared values. We draw chi squared distribution, calling non-symmetrical. Longer tail on the right. And recall that when we read these numbers from a chi square table, we're looking at those probability values to the left of the chi squared value. So, for the right tail, that's going to be the value at alpha divided by 2. So, 0.025 is the probability under that area of the curve. And then for the left tail. That's going to be 1 minus alpha divided by 2. Or sorry, that's confusing the way it was labeled. 1 minus alpha over 2 is going to be the area. On the left side. So let me label that line chi squared. Of 1 minus alpha divided by 2. And the line delineating the right tail. As chi squared of. Alpha divided by 2. So then, since our alpha divided by 2 is 0.025, 1 minus alpha divided by 2 is equal to 0.975. So that left tail value. It's going to be chi squared. At our 1 minus alpha divided by 2. So, in the subscript we have 0.975, and then our degrees of freedom, 14. So look up those values on a chi square table. And the value you will get is 26.119. And then for our right tail. That will be chi squared. It's at alpha divided by 2, which is 0.025, and then 14. When we look up those values on the table, Oh, and here you can see I've mixed this up, and I'm going to leave this, um, leave this mistake in here because this is a really common way to mix things up, and we have a way of thinking through logically and double checking this. So we're looking at that left tail tail value, we would expect that to be smaller. That's the lower tail and reflects a smaller value. So, this should 26 is going to be too large. So I'm going to leave that 26 there, cross it out and write too large. So it's easy to mix these things up and you can kind of do this gut check with these chi square values to tell you that you've reversed them. So, 0.975 and 14 is going to be 5. 629. And then our high scored value for the right tail. At 0.025 with the degrees of freedom 14 is 26.119. That's the larger value. So, let's check our multiple choice options, and we see that this corresponds with. Choice A, which has our left tail value at 5.629, and our right tail value at 26.119. So, once again, we have critical values of chi squad for 1 minus alpha over 2, and chi squad for alpha over 2, for a confidence level of 95%, with the sample size N equals 15. And that is choice A. Chi squad of 1 minus alp over 2 is 5.629, while Chi squad of alp over 2 is 26.119. See you in the next video.

In Problems 5–8, find the critical values χ²₁₋ᵅ⁄₂ and χ²ᵅ⁄₂ for the given level of confidence and sample size.
8. 99% confidence, n = 14

Key Concepts

Chi-Square Distribution

Degrees of Freedom

Critical Values and Confidence Intervals

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