Determining if a set of experimental results is statistically significantly different from zero?

Question

We have a dataset resulting from an experiment with a genotype A and a genotype B, which underwent either treatment X or no treatment Y, for four total sample groups. Each sample group had only three replicates. The measurements are cell counts. In a two-way ANOVA, we found a significant difference between treatment and non-treatment, and did not find a significant difference between the genotypes. That's all good.

The issue is that apparently we expected the non-treatment groups to have values of zero- and they're not. A t-test shows that the difference between groups (A,Y) and (B,Y) to be not significant. I've been requested to determine if the non-zero values of those groups are statistically significantly different from zero or not. While it doesn't seem to make a whole lot of sense to me to do this, I need to provide something, and I'm not sure of an appropriate test that can be done in this case. If I pool all the values for Y, I get a mean of 36, SD of 24 and 95% CI of 19, so the CI doesn't even overlap zero. Is there an acceptable way of answering this statistically, or should I let them know that it's not exactly an appropriate question?

Answer 1

It's an appropriate enough question, and you can answer it with the data you have. If I'm reading your question correctly, you have six Y replicates, so the standard error of the mean of Y is 9.8. Running a two-tailed T-test with five degrees of freedom indicates that values within 2.57 standard errors of the mean are to be found in 95% of cases. To get the confidence interval, you multiply the result of the T-test by the standard error (9.8*2.57 = 25.2), so you can say "the mean of Y is 36 ± 25.2 (95% confidence interval, n=6)".

With a mean of 36, a score of 0 is about 3.67 standard errors away from your sample mean. Looking at the T-table for 5 d.f., we can determine that there is about 0.7% chance of the population mean for Y being as low as 0 given the data you've collected. Since we need to use a two-tailed T-test (we didn't know in advance whether Y would be above or below 0), so we can say that the mean of Y is statistically significantly different from 0 when using the 95% confidence interval, but not when using the 99% confidence interval.