- #1

- 673

- 29

Suppose ##X## ~ N(##\mu##,##\sigma^2##). Then ##\bar{X}## ~ N(##\mu##,##\frac{\sigma^2}{n}##), where ##\bar{X}## is the random variable for sample mean for samples of size ##n##.

But when the population variance ##\sigma^2## is unknown and the sample size ##n## is small, ##\bar{X}## no longer follows a normal distribution but instead follows a t distribution, such that ##T=\frac{\bar{X}-\mu}{S/\sqrt{n}}## ~ t##_{n-1}##, where ##s^2=\frac{n}{n-1}\times##sample variance##=##the unbiased estimator of the population variance and ##n-1## is the degree of freedom of the t distribution.

My question is why does the distribution of ##\bar{X}## changes just because we do not know the population variance? Shouldn't the population variance still be some fixed value ##\sigma^2## (it's just that it's unknown to us at the moment), and thus making ##\bar{X}## follow a normal distribution still: ##\bar{X}## ~ N(##\mu##,##\frac{\sigma^2}{n}##)? It seems that objective reality (the specific distribution of ##\bar{X}##) changes according to subjective knowledge (whether we know ##\sigma^2## or not). And this I find puzzling.

But when the population variance ##\sigma^2## is unknown and the sample size ##n## is small, ##\bar{X}## no longer follows a normal distribution but instead follows a t distribution, such that ##T=\frac{\bar{X}-\mu}{S/\sqrt{n}}## ~ t##_{n-1}##, where ##s^2=\frac{n}{n-1}\times##sample variance##=##the unbiased estimator of the population variance and ##n-1## is the degree of freedom of the t distribution.

My question is why does the distribution of ##\bar{X}## changes just because we do not know the population variance? Shouldn't the population variance still be some fixed value ##\sigma^2## (it's just that it's unknown to us at the moment), and thus making ##\bar{X}## follow a normal distribution still: ##\bar{X}## ~ N(##\mu##,##\frac{\sigma^2}{n}##)? It seems that objective reality (the specific distribution of ##\bar{X}##) changes according to subjective knowledge (whether we know ##\sigma^2## or not). And this I find puzzling.

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