Subscribe Bookmark RSS Feed

t-test

Adias

Occasional Contributor

Joined:

Mar 2, 2017

Hello,

Can I get good performance in the t-test for mean comparisons, even if the normality assumption is violated ? (This in the case of comparison of two populations that have similar number of observations for each one (n = 41 and n = 42). I need some references to support this.

Thank you

Adias 

1 ACCEPTED SOLUTION

Accepted Solutions
markbailey

Staff

Joined:

Jun 23, 2011

Solution

The curvature in the normal quantile plot suggests that there is some skew in the population, one of the reasons that the goodness of fit test rejects the normal distribution model. The skew is not that strong, though, so the sample means are approximately normally distributed after all and the t test should be valid.

Here is a reference for estimating the minimum sample size necessary to assure that the sum of the random variables is normally distributed:

Sugden, R. A., et al. (2002) "Cochran's Rule for Simple Random Sampling,
J of the Royal Statistical Society, Series B, Statistical Methodology. 62(4):787-793.

Learn it once, use it forever!
6 REPLIES
markbailey

Staff

Joined:

Jun 23, 2011

If the populations are not normally distributed, the assumption that the sample means may not be violated if the sample size is large enough. The Central Limit Theorem says that the sum of N random variables is normally distributed for large N. The size N depends on the skewness of your population.

In what way and to what extent are the populations not normal?

Learn it once, use it forever!
Adias

Occasional Contributor

Joined:

Mar 2, 2017

Thank you,

"In what way and to what extent are the populations not normal?"

By plot distribution and Shapiro-Wilk W test (alpha = 0.05). In the figure attached there is an example of the plot and test for one population.

 

 

markbailey

Staff

Joined:

Jun 23, 2011

Solution

The curvature in the normal quantile plot suggests that there is some skew in the population, one of the reasons that the goodness of fit test rejects the normal distribution model. The skew is not that strong, though, so the sample means are approximately normally distributed after all and the t test should be valid.

Here is a reference for estimating the minimum sample size necessary to assure that the sum of the random variables is normally distributed:

Sugden, R. A., et al. (2002) "Cochran's Rule for Simple Random Sampling,
J of the Royal Statistical Society, Series B, Statistical Methodology. 62(4):787-793.

Learn it once, use it forever!
Adias

Occasional Contributor

Joined:

Mar 2, 2017

Thank you Mr Markbailey for your attention!

Peter_Bartell

Joined:

Jun 5, 2014

Without an operational definition of 'good performance' it's impossible to answer your question. If all else fails I suggest any one of the non-parametric tests for testing the hypothesis for two population means. This way you don't have to come up with a definition for 'good performance' and you aren't necessarily tied to any distributional assumptions.

markbailey

Staff

Joined:

Jun 23, 2011

You could also perform the t test with the Oneway platform (Fit Y by X) and then bootstrap the difference with JMP Pro.

Learn it once, use it forever!