You are working with a sample of n=16 selected from an underlying normal population, S=10. What

You are working with a sample of n=16 selected from an...

Key Concepts

Test Statistic for Variance

To test hypotheses about a population variance under the assumption of normality, the test statistic is calculated as (n - 1) times the sample variance divided by the hypothesized variance squared. This statistic follows a chi-square distribution with n - 1 degrees of freedom, providing a basis for assessing whether the sample variance is consistent with the hypothesized population variance.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis while estimating a parameter. In variance testing using sample data, the degrees of freedom are typically equal to the sample size minus one (n - 1), which accounts for the estimation of the sample mean.

Chi-square Distribution

The chi-square distribution is a probability distribution that arises from the sum of the squares of independent standard normal variables. It is widely used in hypothesis testing and confidence interval estimation for variance, particularly in cases involving normally distributed data.

Transcript

0:00 Hi everyone.

00:01 So today we're going to be doing a similar question to the previous video, but we're searching for the population standard deviation rather than looking for the sample mean.

00:10 And so i'm just going to write out the information we're given in the question.

00:13 So we're given hypotheses h -0, mu is equal to 16, h -a, mu is less than 16.

00:25 We're given that it's a sample of 50, so n is 50.

00:30 We're told that x bar is 14 and we are given a p value of 0 .017.

00:42 Now the first thing we need to do is convert this p value into a z star value.

00:47 And we do this using the inverse normal, which i talked a little bit about when i answered question 109.

00:54 So i'd look back to that if you want a little bit of an idea what's going on with this calculation.

00:58 But you basically plug it into a calculator.

01:00 You get it out as 2 .1201.

01:04 So now we've got a z star value and we also have a handy z star formula.

01:08 And so we can use this formula here and the values that we've got available to us to isolate the value of sigma.

01:16 So if i write this z star value here on the left hand side, and write 2 .11211 is equal to.

01:24 And then we know what x bar is because we were told that it was 14 over here.

01:29 And we know what mu is because we were told what mu is.

01:33 And that's 16.

01:35 Now we're looking for sigma and we've been told that n is 50.

01:43 And so we've got sigma.

01:45 Now this is a little bit harder than just isolating the sample mean because this is in the center of a sort of double -decker fraction essentially...

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