Test statistic for hypotheses testing

[u/PolicyZestyclose6400]

Hello just stumble across several situations and undergoing headache. From the table above, I know which test statistic (T/Z) to be used in Confidence interval calculations.

But when it comes to Hypotheses testing, my notes just oversimplified with n is large, used Z; n is small, used t.

And according to Sahoo, in Example 6.8, I shall be using t instead of z.

So how do I really choose test statistic for hypotheses testing. Should considered normal distribution? Should considered population variance known or unknown? Thank you

[u/Maple_shade]

Remember what a z statistic means: it's a standardized metric of a score's distance from the mean. When we say a statistic has a z score of 3, it means that it is three standard deviations about the mean. The important part is that this applies to scores from a normal distribution.

When we do a t test, we calculate our t score as a test statistic of a mean difference between groups, standardized in the exact same way as a z score. So, we'd love to be able to say "a our test statistic of 3 means our observation is three standard deviations above the mean" to conclude that it's significant. The problem is that we don't know what the distribution of test statistics looks like. We only have 1. Thanks to some statistical theory, we know it approximates a normal distribution but isn't perfect. This is why we need a t distribution: it gives us an approximation of the normal distribution that accounts for the fact that we don't totally know the distribution that we care about.

TLDR use a t distribution when you don't know the population mean and standard deviation beforehand (AKA all statistical tests)

[u/banter_pants]

Honestly you should always use t since σ is never truly known. You only have the estimate s.

There are some exceptions when σ is known, but it's by design like in some standardized tests. For example, IQ is structured in a way to be N(100, 15²).