Distribution normality check I can not solve the problem from my homework. 

We conducted two experiments. In the first, there were 400 patients,
  and in the second, 250. In these experiments, the effects of various
  drugs were evaluated. The average weight of people in the two groups
  was compared using the t-test. The test of normality for the first
  group gave a p-value below the significance threshold, and for the
  second above the significance threshold (but the histogram was
  bell-shaped). Variance in groups differed by more than 30%. Is it
  possible to say that the experiments were compared incorrectly?

There are answer answers to choose, only one of them is correct:


*

*everything is bad, the distribution is not normal, we need a Wilcoxon test

*everything is bad, the samples have different variance and sizes, you cannot use the t-test

*the samples are large enough so that everything is fine

*everything is fine, p-value of one of the groups is larger than the threshold, so the result of the first group could be random, then the distribution is still normal


Personally, I think that the correct answer is the second, because the condition for the applicability of the t-test is homogeneity of variance, and in these experiments it is very different. But I'm not sure that this is the right answer.
 A: The question is very poorly constructed and contains some serious flaws. The context of the question may help. Looking at the 'rules' and 'guidelines' for two-sample t tests that have been given just previously to the question may help you figure out what the author means. 
Major flaws are as follows:


*

*"[T]he second [P-value is ] above the significance threshold (but the histogram was bell-shaped)." I agree with @statsandr (+1) that this seems self-contradictory.

*"Variance in groups differed by more than 30%." The appropriate way to judge whether sample variances indicate the population variances may be unequal is to look at their ratio, not their difference.

*Nothing is said about the difference in sample means and no clue is given how large a difference would be of practical importance. So, against what standard are we to judge an "incorrect" comparison? 
Also, we don't know whether the two-sample t test under discussion is a 'pooled' or a 'Welch' test. A Welch test should take care of a difference in variances. The DF of a Welch test can't be below $\min[(n_1 - 1),(n_2 - 1)] = 249,$ so the t statistic must be nearly normal.
If a real-life situation, using a Welch t test, is described here, my guess is that everything is OK. But the exposition of the question is so foggy that my crystal ball doesn't say which answer its author expects.
A: @1. The importance normality decsreases while N increases. See here or here. "With large enough sample sizes (> 30 or 40), the violation of the normality assumption should not cause major problems. This implies that we can use parametric procedures even when the data are not normally distributed" (from first source).
@2. This may be a problem since if sample sizes are unequal, unequal variances can influence the Type 1 error rate of the t-test by either increasing or decreasing the Type 1. Still, you may want run a Levene's test or to see how much the varianes differ.
@3: To my understanding big sample sizes don't guarantee that everything will be fine. Ok, we sad that with increasing sample size the normality assumption becomes less inportant in 1. I am not too sure about the same being true for unequal variance.
@ 4. I don't understand that sentence. And in fact, I find this sentence confusing, too "for the second above the significance threshold (but the histogram was bell-shaped)". Why does it say "but"? A significant test for normality would indicate data not being normal not the other way. At least this is the case for tests I know like the Shapiro-Wilk test where "The null-hypothesis of this test is that the population is normally distributed. Thus, on the one hand, if the p value is less than the chosen alpha level, then the null hypothesis is rejected and there is evidence that the data tested are not normally distributed." (source)
EDIT
Thanks to @Glen_b who pointed out that the quote under "@1" should be restrained because "while the t-test may end up having a nice normal-looking null distribution in many cases if n is large enough, its performance under the null isn't really what people care most about -- it's performance under the alternative -- and there it may not be so great, if you care about rejecting the null in the cases where the effect is not so easy to pick up." (quote from Glen_b)
