Statistics 541: Confidence intervals, prediction intervals, assumptions

Admistrivia

• Homework:
  • Read Myers section 2.7 and do problems 2.21 and 2.24 (due next Tuesday)
  • Run a simple linear regression in both JMP and Splus. Print the output. Circle everything that makes sense to you (i.e. if the Durban-Watson statistic is printed and you haven't any idea what it is, don't circle it!) Most likely, more than 1/2 of the numbers will be mysterious.

SEs for coeficients (page 29-30)

• b₁ is Normal(beta₁, sigma²/S_xx)
• c, the centercept is Normal(beta₀ + beta₁ X-bar, sigma²/n)
• b₁ and c are independent (difficult)
• b₀ is Normal(beta₀, sigma²(1/n + x-bar²/S_xx))

Confidence intervals for observations (section 2.9)

• y-hat = c + b₁ (x - x-bar)
• So standard error is computed in the same ways as for intercept!
• Alternative idea: shift data to make the x of interest be zero
• CI for y is y-hat +/- 2 SEs
• What does this interval forecast?

Prediction intervals for observations

• Suppose we want to predict a future observation Y and we know x
• use above to predict E(Y|x)
• But Y - E(Y|x) has standard deviation sigma
• So prediction - Y has variance = SE(prediction)² + sigma²
• What does this interval forecast?

Draw pictures of CI and PI

• linear bounds
• extrapolation not any where near wide enough
• What if something doesn't lie in the prediction bounds?

Assumptions

• linearity (y = a + bx)
• zero mean errors (duh, how could they be anything else?)
• constant variability (homoskadastic)
• normal distribution for errors
• Independence

Residuals (section 2.12)

• goal: check all of the above assumptions