Statistics 102H: Predictions and CI

Basic "theory"

• Y = fit + noise
• If everything goes well: relatively speaking fit has smaller variability
• Prediction interval is based on SD of noise
• Empirical rule:
  • +/- 1 = 2/3s of data
  • +/- 2 = 95% of data
  • +/- 3 = 99.8% (ivory soap) of the data
• What can go wrong?
  • curvature
  • non-independence
  • hetroskadasticity
  • non-normality
• Hence checking assumptions is necessary to believe intervals

Ignore fit inacuracy

• start by assuming alpha, beta, sigma are perfictly estimated
• What is the guess for y for a new observation? (Y-hat = alpha-hat + beta-hat X.)
• Works if there is a lot of data
• prediction intervals are basically flat
• Try it with idealized regression? (Or use diamond data)

Homework

• Compute the solution to the least squares equation I presented in class today. In other words, find the alpha and beta that minimize:

  Sum(Y_i - alpha - beta X_i)²

• You can start looking at the first short assignment.