Two Tailed and One Tailed T Test in R

Question

Histogram

Dataset Boxplot

Above is the data and some overview of how it looks. I want to use a t test to find if the mean width between 2 samples from the same population are different (2 Tailed). More specifically, I am interested if the mean width from group b is larger than the mean width from group a (1 Tailed).

When using t-test and hypothesis testing in general, I know that you are supposed to define the null hypothesis (and what you expect to see) before you conduct the tests or examine the data. Here the sample is roughly normal, and I will not assume equal variance

Some Questions I still have:

1. I Perform a 2 tailed test and see that there is a statistically significant difference in the 2 means. From here, Could I run a 1 tailed test in both directions to test whether group b has a larger mean width ? What is the issue with performing a 2 tailed test before a 1 tailed ? In my mind that makes sense... Test for a difference, then test for the direction of the difference.

2. If my sample size were much larger for each group, say n = 500 for each, would I need to check the normality of the data ? Or would the number in the sample be sufficient to proceed with non normal data ?

3. Interpreting the confidence level: If we have alpha =.05 and our p-value is less than that, can we say that we are 95% confident that the difference in the mean values are different ?

Answer 1

1. I would only go for one test, usually one uses a two-tailed test. To do a one-sided test, you should have been able to justify it BEFORE looking at your data. For example, if I drop a ball from two different heights, I would believe the time for the ball dropped from a lower height to have a smaller time before hitting the ground. So I could justify using a one-sided test.
2. Generally people apply t-test even if distributions are not normal. Though, you could use non-parametric tests like the sum of rank. Or you could use ANOVA which is fairly robust to deviation from normality.
3. You could say that you are p-value*100 % confident that you did not make a type 1 error when you rejected the null hypothesis. Or, if you performed this experiment 100 times, you would incorrectly reject the null hypothesis p-value*100 times. For me, this is usually the easiest way to think about the interpretation of the p-value.