Which version of the t-test (or other) should I use? UPDATE: I added histrograms and qqplots at the bottom.
A dataset of 1500 scores is close to being normally distributed, but is somewhat pointy, and is right-skewed.  It has a mean of 500, and a standard deviation of 80. (It fails normality tests, such as Shapiro-Wilks an a qqplot.)  The test was given in all 50 states.
Subset A, from Washington, is 100 scores. The histogram shows the distribution is much flatter than a normal distribution, and is right-skewed. (But is still in the bell-shaped class.)
The mean is 600, and the standard deviation is 105.
Subset B, from Oregon, is 140 scores. The histogram shows the distribution is more concentrated around the mean than a normal distribution, and is right-skewed. (But is still in the bell-shaped class.)
The mean is 490, and the standard deviation is 82.
I want to test whether these two "samples" could have been randomly drawn from the nationwide dataset, to show whether there is a significant difference in the states (attributable to state demographics, education level, methodology, etc.). I also want to show whether there is a significance difference between the WA and OR scores.  What tests can I use to show these things?  Can I use Z-scores on these two states?


 A: Ordinarily, this would be an addendum to my previous answer. However, it is as much about my discussion in Comments with @Glen_b as it is about OP's Question.
A discussion about the robustness of t tests against some kinds of
non-normal data often begins by looking at normal probability plots and
investigating loss of power for various non-normal distributional shapes.
(See, recent editions of Ott & Longnecker: Intro. to Stat. Methods and Data Analysis, Ch 5, for one such approach.) Other accounts say that the CLT
may make sample means nearly normal and normality of the data overall
may not be so important. Some of these include bogus statements about
20 or 30 observations being some sort of magic guarantee that a t test
will be robust. Seldom mentioned is @Glen-b's point that $\bar X$ and $S$ need to be independent in order for the t statistic to have a Student's t distribution.
If one is to complain about careless promises of robustness, it seems
a good idea to have at least one specific problematic case in mind.
An exponential sample of size above 30 seemed a likely candidate, so
that is were I begin.
Perhaps the first concrete warning sign is that, for such samples, t tests at the intended 5% level do not actually have significance level 5%. More like 7%.
set.seed(1234)
pv=replicate(10^5, t.test(rexp(35),mu=1)$p.val)
mean(pv < 0.05)
[1] 0.06852

Consider, specifically, this sample y of size $n=35$  with $\bar Y = 0.968$ from $\mathsf{Exp}(1).$
set.seed(1235)
y = rexp(35)
mean(y)
[1] 0.9675814

Multiple t statistics, intended to test $H_0: \mu=1$ vs. $H_1:\mu\ne 1,$ can be
simulated by re-sampling from y as follows:
set.seed(1236)
m = 10^5;  t = numeric(m)
for(i in 1:m) {
  y.re = sample(y,35,rep=T)
  t[i] = (mean(y.re)-mean(y))*sqrt(35)/sd(y.re) }
summary(t)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
-10.02747  -0.89537  -0.07087  -0.27109   0.57515   3.10906 

The distribution of these values t is not distributed as $\mathsf{T}(\nu=35).$ The histogram does match the appropriate t density, and
the ECDF of the first 5000 values does not match the appropriate CDF.

And a Kolmogorov-Smirnov test on the first 5000 ts strongly rejects $\mathsf{T}(\nu=35)$ as the
correct distribution of the purported t statistics.
ks.test(tt, "pt", 34)

        One-sample Kolmogorov-Smirnov test

data:  tt
D = 0.071059, p-value < 2.2e-16
alternative hypothesis: two-sided

The averages $\bar Y$ of samples of size 35 from $\mathsf{Exp}(1)$ are
distributed as $\mathsf{Gamma}(35,35)$ (blue density), not
exactly normal (red dots). More seriously, the sample means and standard
deviations are not independent, so 't statistics' cannot have a t
distribution. (In particular for $X_i > 0,$ one has $\bar Y \ge S/\sqrt{n},$
so no point can lie above the line $S = \sqrt{n}\,\bar Y.)$

set.seed(1237);  m = 50000;  n = 35
v = rexp(m*n);  DAT = matrix(v, nrow=m)
a = rowMeans(DAT);  s = apply(DAT, 1, sd)
par(mfrow=c(1,2))
hdr="Averages of Exponential Sample Not Normal"
 hist(a, prob=T, col="skyblue2", main=hdr)
  curve(dgamma(x,35,35), add=T, col="blue", lwd=2)
  curve(dnorm(x,mean(a),sd(a)), add=T, col="red", 
        lty="dotted", lwd=2)
hdr="Averages and SDs of Exp Samples Not Indep"
 plot(a,s, pch=".",main=hdr)
par(mfrow=c(1,1))

Note: An exact test for means of exponential samples, based on $\mathsf{Gamma}(\mathrm{shape}=35, 
\mathrm{rate}=35),$ is available.
