To test if the means of two or more groups are different from one another.
The ANOVA (analysis of variance) procedure can be used when you compare the means of two or more samples. (In the two-sample case a t-test or the ANOVA procedure can be used. Even though the procedures will be different, the conclusions will be the same.)
First remember that the variance is the square of the standard deviation. Working with variances means not having to take a square root.
The ANOVA procedure allows us to check if the means of groups are significantly different from one another, by calculating their variance. We can do this by treating means like observations, and calculating two different variances. The first variance is the one between means, i.e. the variance of the j 's. If the difference between these averages is large then that variance is going to be big. However, one must defione large based on the random noise in the data. Thus you need a second variance based on the residuals, the eij . If the ratio between these two variances is high (i.e. the differences between the means is high relative to the variance of the residual) then the means must be significantly different from one another.
This ratio of variances is called an F-statistic, and is named after R.A. Fischer, the father of modern statistics.
In all ANOVA tests (even though it is not stated directly) the null hypothesis will be that the F statistic, the standardized variance among the means, is 0. This can only happen if each all the means of the different groups are the same.
Calculating the test statistic from
Do not be intimidated by these expressions. The Mean Sum of Squares for the groups is the variance of the means weighted by the size of the sample. The secon row gives an "average" standard deviation.
Examples (Solutions available)
Run a hypothesis test using the ANOVA
procedure for each of these examples using Minitab if necessary.
The excel files have a template set up which calculates the ANOVA table values.
|1. Given the summary statistics as shown
in the following table. Set up an ANOVA table and test if there is a difference
between the means.
Is the education level of an adult an
indicator as to the ability to remember pictures? Thirty individuals, 10
each from three groups: high school education only, two years of college,
and completed college degree, were given 18 colored pictures to remember.
Four weeks later they were asked questions about the pictures and the number
they remembered was recorded. The results are given in the table
below. Test if education level is an important factor in the ability to
|3. A data set containing information on medium
sized North Carolina counties from the years 1999 - 2000 is available in
Minitab (or Excel)
format. This file contains the County name, its projected population, the
region of the state in which it is located, its property crime rate (incidents
per thousand), average income (in $1,000) of its residents, and the resident's
unemployment rate. (Only counties with populations ranging from 20,000
- 99,000 were considered.) This data is available from NC State data center
Using that data determine whether crime rates are different for different
regions of the sate. Set up the hypotheses, check if the assumptions
are met, and run the hypothesis test. Then answer the question, "Do
property crime rates vary between different regions of the state?"