- Handout for homework

- Need to simulate a whole experiment in our heads
- Suppose we are testing H
_{0}: mu = 0 vs. H_{1}: 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

- 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 - mu
_{0} - Z
_{alpha}= required significance (alpha) - Z
_{beta}= required power (beta) - Key equation: |Z
_{alpha}| + |Z_{alpha}| = delta/SE - n = sigma
^{2}(|Z_{alpha}| + |Z_{alpha}|)^{2}/delta^{2}

- delta = mu - mu

- 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)

- Example: to build a new plant or not
- Now solve for n

Last modified: Tue Apr 3 08:50:07 2001

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