Required sample size for T-test and ANOVA I remember that in using z-test vs t-test, the required sample size for z-test is n>30 while in t-test n<30 (Generally, is this the answer for the maximum sample size for t-test?)
In ANOVA, I know that the groups must be at least two but I don't know how many must be the required sample size. (Generally, is it just like t-test which has a maximum sample size of 30?)
 A: Suppose you have a sample of size $n=40$ from a normal population with estimated $\sigma = 10.$ that you want to use a t test $H_0: \mu = 50$ against $H_a: \mu > 50$ at level $\alpha = 0.05 = 5\%,$ and that you want 80% power to reject if $\mu_a = 52.$ Is $n  = 40$ large enough?
Test for one hypothetical dataset (using R) is shown below. The P-value
$0.024 < 0.05 =5\%$ indicates rejection in this one instance.
set.seed(130)
x = rnorm(40, 52, 10)
t.test(x, mu = 50, alt="greater")

        One Sample t-test

data:  x
t = 2.0415, df = 39, p-value = 0.024
alternative hypothesis: true mean is greater than 50
95 percent confidence interval:
  50.61081      Inf
sample estimates:
 mean of x 
   53.4965 

A simulation of 100,000 such datasets and tests will give us
an idea what the rejection probability is.
set.seed(2022)
pv = replicate(10^5, t.test(rnorm(40,52,10), mu=50, alt="g")$p.val)
mean(pv <= 0.05)
[1] 0.34678

So, the actual power against the alternative value $\mu_a = 52$ is only
about 35%, not the desired 80%. With a little experimentation, we
can see that a sample of size of about $n = 160$ would be required for power 80%
in these circumstances. [More specifically, power $0.808 \pm 0.002.]$
pv = replicate(10^5, t.test(rnorm(160,52,10), mu=50, alt="g")$p.val)
mean(pv <= 0.05)
[1] 0.80837                   # aprx power
2*sd(pv <= 0.05)/sqrt(10^5)
[1] 0.002484384               # aprx 95% CI for margin of sim error

Note: Many statistical software programs have pre-programmed
'power and sample size' procedures. Also, some Internet sites
have useful procedures (unfortunately, others have garbage; be careful). Also, I don't know the level of your
statistics background. If you are ready for a bit of theory
and an introduction to non-central t distributions, you can
look for the relevant equations.
