Hypothesis test for the difference of the difference of means?

[CaptainP]

Hello all,

Consider 3 different samples. Each sample is independent.

1: 23, 25, 21, 22, 21, 22, 20, 23, 19, 22, 19, 21
2: 28, 27, 27, 29, 26, 29, 27, 30, 28, 27, 26, 29
3: 23, 20, 25, 21, 22, 23, 21, 20, 19, 20, 22, 21

I am being asked to test whether μ2 - μ1 = μ3 - μ2 with 99% confidence. How am I meant to perform this test? I know how to test for the difference of means, that's quite simple, but how does one test for the difference of differences?

 

[WWGD]

I think this is usually done with ANOVA http://en.wikipedia.org/wiki/Analysis_of_variance, see the section ANOVA for a single factor and below.

 

[CaptainP]

ANOVA tests whether the means of all samples are the same, so in this case an ANOVA test would test whether μ1 = μ2 = μ3. I need something rather different, testing whether the differences between specific pairs are equal.

 

[WWGD]

Sorry I misread the post.

 

[blue_raver22]

I would approach this problem by first defining the null hypothesis and alternative hypothesis. The null hypothesis would be that there is no difference between the means of the three samples, while the alternative hypothesis would be that there is a difference between the means of the three samples.

Next, I would calculate the mean and standard deviation for each sample, as well as the difference of means between each pair of samples (e.g. μ2 - μ1 and μ3 - μ2). This would give me a better understanding of the data and help me determine if there is a significant difference between the means.

To test for the difference of differences, I would use a two-sample t-test. This test compares the means of two independent samples and determines if there is a significant difference between them. In this case, we would be comparing the difference of means between each pair of samples (e.g. μ2 - μ1 and μ3 - μ2) to see if they are significantly different from each other.

To perform the t-test, I would calculate the t-statistic and compare it to the critical value from a t-distribution table. If the t-statistic is greater than the critical value, we can reject the null hypothesis and conclude that there is a significant difference between the means of the three samples. If the t-statistic is less than the critical value, we fail to reject the null hypothesis and cannot conclude that there is a significant difference between the means.

Finally, to determine the confidence level, I would use the t-distribution table to find the critical value for a two-tailed test with 99% confidence. This would give us a cutoff point for the t-statistic, and if our calculated t-statistic falls beyond this cutoff point, we can reject the null hypothesis with 99% confidence.

In summary, to test for the difference of differences in means, we can use a two-sample t-test and compare the calculated t-statistic to a critical value from a t-distribution table. This will help us determine if there is a significant difference between the means of the three samples with 99% confidence.