This tutorial will cover tests for more than two population means and the process for analysis of variance. You're going to learn about:
What is a circumstance where you might see an ANOVA scenario?
A factory supervisor wants to know whether it takes his workers different amounts of time to complete the task based on their proficiency level. The factory employs apprentices, novices, and masters. The supervisor randomly selects 10 workers from each group and has them perform the task. The summary of the data, time in minutes to complete the task, is shown in this table here:
Are these sample means significantly different from each other?
In order to answer this question, you will need to perform the analysis of variance.
Analysis of Variance (ANOVA)
A hypothesis test that allows us to compare three or more population means.
Comparing three or more means requires a new hypothesis test called analysis of variance or, more typically, ANOVA (the AN is for "analysis", the O is for "of", and the VA is for "variance").
For ANOVA, compare the means by analyzing the sample variances from the independently selected sample. There are a few conditions necessary for this test:
For our scenario, assume that these conditions are met.
In an ANOVA test, it's important to identify the null and alternative hypotheses.
For our scenario:
When you do an ANOVA test, the statistic that you use is not going to be a z or t, as you have been using in the past. Instead, you will use what is called an F. An F statistic is calculated by taking the quotient of the variability between the samples and the variability within each sample.
F statistic
The test statistic in an ANOVA test. It is the ratio of the variability between the samples to the variability within each sample. If the null hypothesis is true, the F statistic will probably be small.
So a large F provides evidence against the null hypothesis, versus a small F serves mainly to uphold the null hypothesis. You wouldn't reject it if F was small.
A small F, once again, is consistent with the null hypothesis, versus a large F statistic is evidence against the null hypothesis.
Almost always, you will calculate the ANOVA F statistic and the p-value with technology. All but the most simple, straightforward problems will be calculated on technology.
In our scenario, the F statistic, calculated with technology, is 1.418. That is not a very large value of F. And the p-value is 0.26, which is a very large p-value.
Since the p-value is very large, greater than the 0.05 significance level, you fail to reject the null hypothesis. There's no evidence that suggests that the time required to complete the task differs significantly with proficiency level.
ANOVA allows you to compare three or more means by comparing the variability within each sample to the variability between the samples. The null hypothesis is that all the means are the same and the alternative hypothesis is that at least one of them is different. The F and the p-value are almost always calculated on technology.
Thank you and good luck!
Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS
A hypothesis test that allows us to compare three or more population means.
The test statistic in an ANOVA test. It is the ratio of the variability between the samples to the variability within each sample. If the null hypothesis is true, the F statistic will probably be small.