Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — a probability for a sample mean
A population has μ=100 and σ=15. For samples of size 25, find P(Xˉ>103).
- 1Xˉ has mean 100 and standard error nσ.
- 2SE =2515=515=3Five times tighter than individual values.
- 3z=3103−100=1Standardise using the standard error, not σ.
- 4P(Z>1)=1−0.8413
- 5=0.1587About one sample in six would have a mean above 103.
Example 2 — a confidence interval
A sample of 64 items has mean 50, from a population with σ=8. Find 95% and 99% confidence intervals for μ.
- 1SE =648=1
- 2For 95% confidence, z=1.96.Leaving 2.5% in each tail.
- 350±1.96(1)=(48.04, 51.96)
- 4For 99% confidence, z=2.576.
- 550±2.576(1)=(47.42, 52.58)Wider — more confidence requires more room.
- 6Interpretation: about 95% of intervals built this way would contain μ.Not: '95% probability that μ is in THIS interval'.
Example 3 — choosing a sample size
With σ=8, how large a sample gives a margin of error of 1 at 95% confidence?
- 1The margin of error is E=znσ.Half the interval's width.
- 2Rearranging: n=(Ezσ)2
- 3=(11.96×8)2=(15.68)2
- 4=245.9
- 5Round UP to n=246.Rounding down would leave the margin above 1, failing the requirement.