Use normal sampling distributions and standard errors

Move from a probability model to a null model

The sampling examples supply the tail calculation and standard error. A hypothesis test additionally fixes a null reference and a prespecified alternative. For the model below, take 16 independent observations from a normal population with mean 18 and variance 36

Normal inputs give exact normal means

For independent observations from the same normal population, the sample mean is exactly normal for every positive sample size. Its standard deviation is called the standard error. This follows from the independent normal combination rule: dividing the sum by the sample size divides its variance by the square of that size

For 16 independent observations from a normal population with mean 18 and variance 36, give the standard error

Compute a tail using the standard error

For X̄~N(18,9/4), Pr(X̄>21)=Pr(Z>2)=1-Φ(2)≈0.0228. For 16 independent observations from a normal population with mean 18 and variance 36, using the individual standard deviation of 6 would instead answer the question about an individual observation exceeding 21. State whether normality is exact from normal inputs or approximate under a justified large-sample argument

Estimate an unknown standard error

When σ is unknown, use the observed sample SD s to estimate it. The estimated SE is s/√n, not s/n. For n=49 and s=7/2, estimated SE=1/2 and estimated variance of the mean=1/4. These estimates vary from sample to sample. Replacing σ by s does not turn an arbitrary small-sample population into a known normal model

Give the estimated standard error for n=49 and s=7/2.

Large-sample normality is an approximation

For iid observations with finite positive variance, the standardised sample mean approaches a standard normal distribution as n increases. The usefulness of a finite-n approximation depends on population shape and tails. A sample size of 30 does not universally guarantee normality. Increasing the repeat count R improves the empirical display of a fixed-n distribution; it does not change that distribution

Carry the direction and assumptions forward

A correct normal probability is not by itself a test conclusion. Preserve the parameter, hypotheses, significance rule and sampling assumptions