Using the Levene-Brown-Forsythe test for standard deviation hypothesis testing

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to use the Levene-Brown-Forsythe test for standard deviation hypothesis testing.  Here's our problem statement: The accompanying data table includes weights in grams of a simple random sample of 40 quarters made before 1964 and weights of 40 quarters made after 1964.  When designing coin vending machines, the standard deviations of pre-1964 quarters and post-1964 quarters must be considered.  Use the Levene-Brown-Foresythe test and a 5% significance level to test the claim that the weights of pre-1964 quarters and the weights of post-1964 quarters are from populations with the same standard deviation.

Part 1

OK, the first part of this problem asks us for the null and alternative hypotheses.  We're asked to let the weights of pre-1964 quarters be Population 1 and the weights of post-1964 quarters to represent Population number 2.  Well, this is pretty simple with the null alternative hypothesis.  The null is always a statement of equality.  And we see that among our three options here, we have that same null hypothesis for each answer option.  So what distinguishes them is the alternative hypothesis.  We're testing the claim that the two populations have the same standard deviation.  So that means that the standard deviation is equal and also the variance is equal, but equality belongs to the null hypothesis by definition.  So the alternative hypothesis is going to have to come from the complement of our claim, which is that the two are not equal to each other.  So we want to select Answer option A.  Excellent!

Part 2

And now the second part of this problem asks for the test statistic.  And this is the beginning of many, many struggles for students who want to pull their hair out (assuming they have any hair on their head) because this just frustrates them to no end.  They see that we're testing standard deviation.  So of course they want to run the variance testing through StatCrunch.  But notice why that doesn't work.  We're asked for our test statistic, which is a t-score.  See the T down here.  There's no Chi-squared, it's a T.  And this is why students are getting this part of the problem incorrect consistently; it's because they're not using the right procedure.

Here in the problem statement it says, "Use the Levene-Brown-Forsythe test."  So what is the Levene-Brown-Foresythe test?  Well, that's a test where you transform the data using the median value of each of your samples.  And then with the transformed data you perform an independent t-test.  So that's we're going to do to get our test statistic here for this second part.

Of course the first step to do that is to dump the data into StatCrunch.  So here we can dump the data into StatCrunch.  I'm going to resize this window so we can see everything just a bit better.  OK, now here in StatCrunch, the first thing we need to do is transform our data.  And when you're transforming data in StatCrunch, you use the menu option Data --> Compute --> Expression.  So now here in my expression window, these things get really picky.  Computers are very detail oriented, and I'm prone to error.  So I always like to come over here and press Build.  I could just type it in, but again, I feel like I'm too prone to error.  I'm just going to press Build.

And then the transformation for the data is to take the absolute value of the difference between each individual data value and the median value.  So to do that, the first thing I want to do is put it in the absolute value function, which is the first function listed here in the functions list.  So I select that function and then press Add Function.  Now I want to take --- we're going to transform the first data set.  So I take the column for the first --- the first column for the data, press Add Column, and I want to subtract from that the median.  Now, the median is another function here in the functions list.  So I'm going to scroll down here.  So I get to median, I'll select the median, add it to my function, and then I want to select again the same column of data.  So it takes the median for that column and subtracts that from each value in the column, then takes the absolute value.  I click OK, I click Compute!, and here's a new column here where I have my actual transformed data.

I need to perform the same series of steps to transform the second sample of data.  So again, I'm going to go to Data --> Compute --> Expression.  I press Build, and I'm going to go through the same steps that I went through before to build the function that's going to transform that second column of data.  Now I've got the transformation on my second column.

And I'm going to take these two columns of transformed data, and I'm going to perform an independent t-test.  To do that, I go to Stat --> T Stats --> Two Sample --> With Data (because I have actual data here in StatCrunch).  The first column of transformed data is my first sample.  The second is the second, and we want to make sure that this inequality sign matches the one from our alternative hypothesis.  And we see that it does match.  So now we're all set.  I hit Compute!, and here's my test statistic --- the T-stat, second to last value in that table there.  So when I put that value there in my answer field, I'm asked to round to three decimal places.  I check my answer.  Well done!

Part 3

And now the next part asks for the P-value, which of course is always right next door to the test statistic, the last value there in that results window table.  And I'm asked to round to four decimal places.  Conveniently, that's the number I have there in the results window.  Excellent!

Part 4

And now this last part of the problem asks, "What is the conclusion for this test?"  Well, my P-value is a little more than 4%.  We're comparing that with a 5% significance level.  So the P-value is less than the significance level, which means we're inside the region of rejection.  So I'm going to reject the null hypothesis.  And every time I reject the null hypothesis, there is always sufficient evidence.  Fantastic!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below, 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.