Confidence Interval

1. Take a sample from a population
2. Calculate the sample mean
3. Construct $100(1-\alpha )\%$ confidence interval
   1. For a sufficiently large sample, the sample mean follows normal distribution, see (5.4 Central Limit Theorem (CLT))
   2. Use standard normal variable transformation $\frac{\bar{X}-\mu }{\sigma /\sqrt{n}}\approx N(0,1)$
   3. CI: $\bar{x}\pm z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$
      Interpretation

If we take many independent random samples from a population, and construct many 90% confidence intervals, then we expect 90% of the intervals to contain the true population mean.