Find the left and right χ2\chi^2-values for a 99% confidence interval with a sample size of 25.

χ L 2 = \chi_{L}^2= 9.89 ; χ R 2 = \chi_{R}^2=

Find the left and right χ 2 \chi^2 -values for a 99% confidence interval with a sample size of 25.

Alright. So now that we've gotten a lot of practice finding chi squared values with our tables, let's go ahead and take a look at one of the most direct applications of this, which is in confidence intervals. Our practice problem here tells us to find the left and right chi squared values. So we're going to be finding two chi squared values for a 99% confidence interval with a sample size of 25. We've done a problem like this before. Go ahead and pause the video if you haven't already and see if you can work this out on your own. Alright? Now remember, this confidence interval or confidence level tells us our alpha. That's what we're going to need to get our chi squared values. Right? So this is our c, which is the 99%. We use the c to get alpha, which is just one minus that confidence interval or level expressed as a decimal. So we just do one minus 99%, which is just going to be 0.01. Now Now if you remember, the reason we need this is because we need to split that area into two. The basic idea, if you remember, is that we have this chi square distribution, which is kind of sort of asymmetric. It kinda looks like this, and we're interested in two values. This is our chi squared on the right. This is our chi squared on the left. Now the area to the right of this critical value, which stretches all the way out to positive infinity, is alpha over two. Now what happens is when we use our chi squared table, we need areas to the right. So when it comes to the left critical value over here, we need to find the area to the right of this value, which is going to be one minus alpha over two. That's why we need one minus alpha over two. Alright, so our alpha over two is just take this number and cut it in half, which is 0.005, and then we take one minus that number, which is gonna be 0.995. Alright. Our degrees of freedom, as a reminder, is just n minus one. We have a sample size of 25. That's n. So our degrees of freedom, it just ends up being 24. Alright. We're going to use these two values along with our degrees of freedom to find these two critical values. Let's go ahead and actually do that in our table. So we again, we just could go ahead and look at locate the two numbers on the first row, point o o five and point nine nine five, and just go all the way down until you hit our degrees of freedom of 24. If you scroll, you should see that it's right about over here. So you got our first value of 9.886, and then our second value of 45.559. Remember, these tables or these values over here are generally for chi squareds on the right, and these are for chi squareds on the left. Alright? So we have these two numbers right over here. Let's go back to our problem here. We're ready to just go ahead and write them in. So again, I usually write these, on the by rounding them to the nearest hundredths place, which is going to be a chi squared on the right of 45.56. And And then over here, we have 9.89. Alright? So, again, those are our two values, and that's really all there is to it. So let me know if you had any any questions. Great job, and I'll see you in the next one.

Find the left and right χ 2 \chi^2 -values for a 99% confidence interval with a sample size of 25.

χ L 2 = \chi_{L}^2= 9.89 ; χ R 2 = \chi_{R}^2=

χ L 2 = \chi_{L}^2= 45.56; χ R 2 = \chi_{R}^2= χ R 2 = 9.89

χ L 2 = \chi_{L}^2= 10.86 ; χ R 2 = \chi_{R}^2= 42.98

χ L 2 = \chi_{L}^2= 10.52 ; χ R 2 = \chi_{R}^2= 46.93

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8. Sampling Distributions & Confidence Intervals: Proportion

Chi Square Distribution

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8. Sampling Distributions & Confidence Intervals: Proportion

Chi Square Distribution

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Multiple Choice

Find the left and right $χ^{2}$ -values for a 99% confidence interval with a sample size of 25.

$χ_{L}^{2} =$ 9.89 ; $χ_{R}^{2} =$

$χ_{L}^{2} =$ 45.56; $\chi_{R}^2=$ 9.89

$χ_{L}^{2} =$ 10.86 ; $χ_{R}^{2} =$ 42.98

$χ_{L}^{2} =$ 10.52 ; $χ_{R}^{2} =$ 46.93

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Verified step by step guidance

Identify the significance level $ \alpha $ for a 99% confidence interval. Since the confidence level is 99%, $ \alpha = 1 - 0.99 = 0.01 $.

Calculate the degrees of freedom $ df $ for the chi-square distribution. For a sample size $ n = 25 $, the degrees of freedom is $ df = n - 1 = 24 $.

Determine the critical values for the chi-square distribution that correspond to the left and right tails of the confidence interval. These are found using $ \frac{\alpha}{2} $ and $ 1 - \frac{\alpha}{2} $. Specifically, calculate $ \frac{\alpha}{2} = 0.005 $ and $ 1 - \frac{\alpha}{2} = 0.995 $.

Use a chi-square distribution table or statistical software to find the chi-square values $ \chi_{R}^2 $ and $ \chi_{L}^2 $ corresponding to the probabilities $ 0.005 $ and $ 0.995 $ respectively, with $ df = 24 $.

Interpret these chi-square values as the right and left critical values for the 99% confidence interval. These values define the interval boundaries for the chi-square distribution.