If a sample is normally distributed, is its population always normally distributed? I know this is correct: "If a population is normally distributed, a sample randomly chosen from it must be normally distributed." For instance, If the sizes of all apples of my farm are normally distributed, 50 apples chose from it should be normally distributed unless randomness was violated.
But, is the following as well correct? "If a sample randomly chosen from a certain population is normally distributed, the population must be normally distributed." That is, can I believe that its population is normally distributed, if I find a certain sample is normally distributed?

If you have, say, an i.i.d. sample, and you compute certain characteristics of it, like skewness and excess kurtosis, and they are close to the Normal's such, and you perform Normality tests that do not reject Normality, then you are reasonably justified in proceeding with the working hypothesis that the distribution of the population from which the sample came, follows a Normal distribution.
But don't try to coat this with more philosophical certainty than that.
