Finding degrees of freedom, critical values, and a standard deviation confidence interval estimate

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find degrees of freedom, critical values, and a confidence interval estimate for standard deviation.  Here's our problem statement: Use that given information to find the number of degrees of freedom, the critical values, and the confidence interval estimate of sigma (the population standard deviation).  It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution.

Part 1

OK, here it says we need a 99% confidence level on our interval.  Our sample size is 25, and our sample standard deviation 0.26.  Using these values, we can actually give everything that we need to give to answer the questions in the problem.  So the first part is asking us for degrees of freedom.  Degrees of freedom is simply one less than the sample size. Here our sample size is 25, so our degrees of freedom will be 24.  Well done!

Part 2

Now the next part asks us for the first of two critical values.  The one on the left is what's being asked for here.  To get the critical values out, I'm going to load up StatCrunch and access the distribution calculator that is to be found inside StatCrunch.  Let's pop that window out, and let's resize it so we can get a better look at what's going on here.

So to access the distribution calculator, I need to go to Stat --> Calculators --> Chi-squared.  I know I need the chi-squared calculator because, if I look here at the statistic I'm being asked to calculate, this is chi-squared.  So I know I need the Chi-squared distribution.  There are two that we're looking for, and so I'm going to have to use the Between option.  Degrees of freedom are what we calculated in the first part.  That's 24.  Here in these spaces are where our critical values are going to show up.  So what we need to put in here is the area in between the critical values, and that's the size of the confidence level, which in this case is 99%.  So I put 99% in, I press Compute, and here we've got our two critical values.  The one on the left (which is the subscript L) is the one we want first.  And I'm asked around to three decimal places.  Fantastic!

Part 3

And now I'm asked for the other critical value, the one on the right.  You can see that with the subscript R.  And we can just take that here from the calculator.  Fantastic!

Part 4

And now the last part asks for a confidence interval estimate on the population standard deviation.  To do that, I'm going to come back here to StatCrunch, and I'm going to go to Stat --> Variance Stats (because that's how we calculate anything with standard deviation inside the StatCrunch application) --> One Sample (because I have only one sample) --> With Summary (because I don't have actual data).

Here in the options window, I'm asked to calculate the sample variance.  Well, we're not given variance here, but we are given sample standard deviation.  If we remember that variance is simply the standard deviation squared, I can get the sample variance I need by squaring the sample standard deviation.  So if I take out my calculator here, I put in that sample standard deviation, and I square it.  That gives me this sample variance.  So now I can put that number here in this field in my options window.  The sample size we're given here in the problem. I want to flip this button for confidence interval, and I want to make sure that the level matches what's asked for in the problem, which is 99%.  I hit Compute!, e voila!  Here are the lower and upper limits for my confidence interval estimate.

But this is the confidence limit on variance.  Notice this is variance.  We need the confidence interval estimate for standard deviation.  So I have to take these numbers here, and I have to take the square root of variance in order to get standard deviation.  So if I just select that and copy it, and I come over here, and I can paste that in, take the square root, and that's my lower limit that I need to put in here.  I want two decimal places.  That would be 0.19.  And the upper limit --- I want to do the same thing with the upper limit.  Copy that out, come here, right click, paste that in, take the square root. There's my upper limit.  Excellent!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos  or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.