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.

6. 95% confidence, n = 25

Accepted Answer

Hi everyone and welcome back. Our next question says, find the critical values I squared sub 1 minus alpha divided by 2. And chi squared sub alpha divided by 2. For a confidence level. Of 90%. With sample size N equals 18. Sorry, my pen's being a little temperamental there. And then we have 4 answer choices with different values. And rather than reading out the long chi squared um description for each one, let me read the two values. I mean, the first one is for 1 minus alp over 2, and the second one is alp over 2. So, for choice A, it's 26.119. And 7.261. For B, it's 8.672, and 9.390. C is 8.672, and 27.587. D is 24.996 and 6.262. So let's think about what we're looking for here. We're given a confidence level. And we need to have an alpha value. When our confidence level is 90%, alpha will equal 1 minus the confidence level. So 1 minus 0.90 and will equal 0.10. And we're talking about um a two-tailed chi squared volume, because we're looking for this alpha divided by 2 and 1 minus alpha divided by 2. So, let's look at alpha divided by 2 is equal to 0.05, and 1 minus alpha divided by 2 is equal to 0.95. And if we think briefly about what that means in a chi squared distribution, remember that that curve is not symmetrical, it's not a normal distribution. So we draw that chi square distribution with the longer right tail. And the two tails we're looking for is we've got our line on the right. And then our line on the left, and recall that chi squared values give us, or I should say represent that area to the right. Of our chi squared volume. So the 1 minus alpha over 2, that's 0.95, represents the lower tail or left tail. And then the. Alpha over 2 represents the right tail. And since it's not symmetrical, we can't assume that these are the same areas. It's not like in a normal distribution where you can have a positive and negative Z score. So last thing we need. For our chi square table is the degrees of freedom. And we recall that degrees of freedom is equal to N minus 1, so our N value is 18, so we have 17 degrees of freedom. So, for our right tail, we have a chi squared value, or we'll do left tail first since that's how our answers are listed. So, for our left tail, Sorry I keep having trouble. With the tease their left tail. That's going to be our chi squared value. At 0. 95. That being 1 minus alpha divided by 2. 17, that's how we note that. And so when we look up that on a chi square table, again, degrees of freedom 17. That gives us that critical value of 8.672. And then for our right tail or upper tail. That chi squared value, um, down as a subscript. we have 0.05 and 17. When we look that up on our chi square table, we get 27.587. And we can see looking at our multiple choice options, um, we only have two that have that 8.672 value, and that's B and C, but then only C has the correct 27.587. So, choice C is our correct answer. So our critical values for chi squad of 1 minus alp over 2, and chi of alp over 2, for a confidence level of 90% with a sample size of 18. Is C, that our chi squad of 1 minus 2 is 8.672. Our chi squad of alpha 2 is 27.587. 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.
6. 95% confidence, n = 25

Key Concepts

Chi-Square Distribution

Critical Values in Chi-Square Tests

Confidence Interval for Variance

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