Intro to Inference

The big picture

• We wish to understand the quality of parameter estimates.

Concepts and tools

• Sample mean
• Standard error
• Confidence intervals
• Hypotheses testing
• The central limit theorem

Example: The sample mean

• Choose a random sample of 100 peope from a large population.
• Compute the mean of the sample, say: \[\bar{x} = \frac{1.50+1.78+\cdots+1.70}{100} = 1.697\]
• Key question: How close is the sample mean \(\bar{x}\) to the population mean \(\mu\)?

Comments

• The estimation of the population estimate with a single point (the sample mean) is an example of a point estimate.
• If we compute a second sample mean, we expect to get a different result.
• Thus, point estimates are not exact.

Standard error

• The standard deviation associated with an estimate is called the standard error, SE.
• Intuitively, this describes the typical error or uncertainty associated with the estimate.
• One can prove that \[SE = \sigma/\sqrt{n}.\]
• We usually use the sample \(\sigma\) in this computation.

Confidence intervals

• If \(\bar{x}\) is a sample mean, we call \[\bar{x} \pm 2\times SE\] a 95% confidence interval.
• That is, if we construct 100 such intervals, we expect that 95 of them contain the actual mean.