Statistics 541: Sample size calculations

Admistrivia

• Handout for homework

Sample size calculations

Simulating an experiment

• Need to simulate a whole experiment in our heads
• Suppose we are testing H₀: mu = 0 vs. H₁: mu > 0
• Collect n samples, compute X-bar, SD(X-bar) = sigma/sqrt(n)
• Draw pictures under null and under alternative
• Rejection region defines alpha
• Acceptance region defines beta under alternative
• Goal:
  • want mu to be small so we can detect subtle differences
  • want alpha to be small so no type I errors
  • want beta to be small so no type II errors
  • want n to be small so we don't have to collect much data
• Problem: ALL of these are playing agisnst each other

Picking n

• Alpha says how far away x-bar has to be to be significant
• Beta for some delta says how far away x-bar has to be from interesting point
• Draw picture on Z-scale
• Put "real" scale underneath it
• n determines ratio of two scales
• Ah, now the details begin
  • delta = mu - mu₀
  • Z_alpha = required significance (alpha)
  • Z_beta = required power (beta)
  • Key equation: |Z_alpha| + |Z_alpha| = delta/SE
  • n = sigma² (|Z_alpha| + |Z_alpha|)²/delta²

Picking all values

• Draw up table of type I and type II error
• specify delta via economics (smallest profitable difference)
• put costs on each error cell
• If null is true, want small alpha, if alternative is true want small beta, unfortunately, this would be self-confirming EVEN if it weren't correct.
• Scientific tradition: alpha = .05, beta = .1 or beta = .2
• economic decisions are more difficult
  • Example: to build a new plant or not
    • if everything works, small profits
    • if not enough new jobs, large loss
    • Must protect null (alpha very small)
  • Example: eliminate night shift to cut down on breakdowns
    • if alternative true, big wins
    • if alternative shown untrue, we missed a few overtime jobs
    • more symetric (alpha approximately = beta, both not too small)
• Now solve for n