How do you know from what population a sample comes? My question is say you take a sample say of people weight and you get in pounds right [100,220,230,240] many people say this sample come from a normal distribution population.
But my question is from a sample how does you tell that it belong to a certain population???
 A: Suppose you have little information, from the sample or the context in which it was sampled, about nature of the population. Then you might want to try one of several kinds of nonparametric bootstrap confidence intervals.
Suppose you have $n = 20$ observations in a vector x in R
with summary statistics and boxplot as shown below.
summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   3.00   13.50   25.50   30.00   51.25   60.00 
boxplot(x, col="skyblue2", horizontal = T)


The data are moderately right-skewed. One indication is that the median is closer to the minimum than to the maximum. Maybe the skewness is a little more than we would expect from normal data. Also, points in a normal probability plot seem more curved than linear.
qqnorm(x);  qqline(x, col="blue")


Assume normal data, use t confidence interval. If we are willing to guess that the data were sampled from a normal population, then a 95% t confidence interval for the population mean $\mu$ [from the procedure t.test in R] is $(20.4, 39.6).$
t.test(x)$conf.int
[1] 20.41577 39.58423
attr(,"conf.level")
[1] 0.95

Nonparametric bootstrap confidence intervals. This is not the place for a detailed explanation of bootstrapping. But the general idea is to take a moderately large number of 're-samples'
of size $n=20$ with replacement from x in order to assess
the variability of the mean of x and thus to make a 95% nonparametric bootstrap. (Here, the word nonparametric means that we have
made no assumptions about the shape of the population distribution. We do assume that the population distribution has a mean $\mu.)$ One simple style of a bootstrap CI (shown below) gives the interval $(21.5, 38.6).$
For many practical purposes it would not make much difference which
95% CI is used. [Because re-sampling is a random process, results of the bootstrap CI differ slightly from one run to the next---unless the same seed is set. Two additional runs gave $(21.4, 38.2)$ and $(21.1, 38.5)$.]
set.seed(1234)
a = replicate(2000, mean(sample(x,20,rep=T)))
quantile(a, c(.025, .975))
    2.5%    97.5% 
21.49875 38.55375 
hdr = "Bootstrap Dist'n of Sample Means"
hist(a, prob=T, col="skyblue2", main=hdr)
 abline(v = c(21.50,38.55), col="red", 
        lwd=2, lty="dotted")


Note: There are better styles of bootstrap CIs--- especially if the bootstrap distribution shown above were markedly skewed. Another style of bootstrap results from re-sampling deviations from the sample mean, as shown below. The
result is $(21.4, 38.9);$ again not very different from
the previous CIs.
a.obs = mean(x)
set.seed(614)
d = replicate(2000, mean(sample(x,20,rep=T))-a.obs)
LU = quantile(d, c(.975, .025))
a.obs - LU
   97.5%     2.5% 
21.45000 38.90125 


Note on data: The data used in the discussion above were sampled in R
as shown below. If we knew that the population distribution is gamma, then a parametric CI based on the particulars
of that family would be preferable. The true population mean is $\mu = 30.$
set.seed(2021)
x = round(rgamma(20, 3, 1/10))

