To apply the f-test, Joe Schmuller defines a hypothesis testing of two population variances. Specifically, Joe describes why you should to use ratio of sample variances to identify the F-Ration, calculate an F-Test and analyze F-Distribution. Joe uses a widget production example to test hypotheses about population variences.
- [Narrator] Now we'll test hypotheses about two variances. When we test hypotheses about two population variances, we use the sample variances. Unlike hypothesis tests about means, we don't subtract one variance from another. Instead, we calculate the ratio of the two variances. The sampling distribution of the ratio of variances is a family of distributions called F, named after Sir Ronald Fisher, a famous statistician. Members of the family are distinguished by the degrees of freedom for sample variance one, is N one minus one, and the degrees of freedom for sample variance two, is N two minus one.
F-Ratio is the name we give to the ratio of the two variances. F-Test is the statistical test that uses the F-Ratio to test hypothesis about population variances. Use an example. Manufacturing firm has to decide whether or not to buy a new widget-producing machine. The firm wants to decrease the variance of the widget length. They plan to compare a sample of 12 widgets from their existing machine with a sample of 12 widgets from a new machine. Here are the hypotheses and alpha.
The null hypothesis is that variance one is less than or equal to variance two. Machine one being the existing machine and machine two being the new machine. The alternative hypothesis is that variance one is greater than variance two and alpha equals point o five. Another way to express each hypothesis is as a ratio. The null hypothesis, the ratio is less than or equal to one and in the alternative hypothesis, the ratio is greater than one. And the data, the sample of 12 widgets from the existing machine, machine one, has a variance of point seven five square inches.
The sample of twelve widgets from the prospective new machine, machine two, has a variance of point four square inches. Do you reject the null hypothesis? To calculate the F-Ratio, here's the formula. By convention, these types of problems, we put the larger variance in the numerator and here's the result. The F-Ratio is one point eight seven five. The decision? With degrees of freedom one equal 11 and degrees of freedom two equal 11 and alpha equals point o five, the critical value is two point eight two.
You can determine this by looking at a table of the F-Distribution or with an Excel function that I'll show you in a moment. The result, one point eight seven five, is not in the rejection region. So, do not reject the null hypothesis. In this spreadsheet, we'll calculate a critical value for F. Cell A two holds the alpha level. B two has the degrees of freedom for sample one and C two has the degrees of freedom for sample two. The critical value will go into cell D two.
So, click in cell D two and from the statistical functions menu, Select F.INV.RT, RT standing for right tail. With the probability box active, select cell A two to enter alpha. With the degrees of freedom one box active, select cell B two to enter the degrees of freedom for sample variance one. Now with the degrees of freedom two box active, select cell C two to enter the degrees of freedom for sample variance two.
Click on OK and the critical value appears in cell D two. In summation, when testing hypothesis about two variances, use the ratio of sample variances. The sampling distribution of the ratio is the F-Distribution. You find the critical value for alpha and if the ratio is in the rejection region, reject the null hypothesis.
He explains how to organize and present data and how to draw conclusions from data using Excel's functions, calculations, and charts, as well as the free and powerful Excel Analysis ToolPak. The objective is for the learner to fully understand and apply statistical concepts—not to just blindly use a specific statistical test for a particular type of data set. Joseph uses Excel as a teaching tool to illustrate the concepts and increase understanding, but all you need is a basic understanding of algebra to follow along.
- Understanding data types and variables
- Calculating probability
- Understanding mean, median, and mode
- Calculating variability
- Organizing and graphing distributions
- Sampling distributions
- Making estimations
- Testing hypothesis: mean testing, z- and t-testing, and more
- Analyzing variance
- Performing repeated measure testing
- Understanding correlation and regression