Stat 301 – Day 21 Adjusted Wald Intervals Power

Last Time – Confidence Interval for  When goal is to estimate the value of the population proportion  General form: estimate + margin of error If the Central Limit Theorem applies (large enough sample size), then an approximate C% confidence interval for  is: Margin-of-error “Critical value”: Determined by confidence level (p. 326) Wider with larger confidence levels Narrower with larger sample size “Standard error”

“Confidence” If repeatedly randomly sample from the population (with same sample size), then in the long run, roughly 95% of intervals will succeed in capturing the population proportion.

PP (p. 336) 1. Let  represent the proportion of all voters who were planning to vote for Landon 2. Technical conditions: n large enough for CLT  Use the sample proportions here too… 3. Calculation 4. Interpretation: I’m 99.9% confident that between 56.9 and 57.1% of voters planned to vote for Landon?

Last Time – Confidence Intervals Have to have a random sample! Gallup poll ExampleExample

Quiz 17

Investigation (p. 331) If given the choice, which would you prefer to hear first – good news or bad news? Descriptive statistics Inferential statistics  Let  represent the proportion of all Cal Poly students who prefer to hear bad news first  95% confidence interval for   Valid?

Investigation (p. 331)

Back to Inv (p. 315) H 0 :  = 2/3 vs. H a :  ≠ 2/3 We failed to reject 2/3 as a plausible value for the population parameter. H 0 :  =.5 vs. H a :  >.5 (most turn right) We would reject.5 (one-sided p-value =.001) Is it possible we are wrong in either case?

Type I/Type II Errors If we reject the null hypothesis, but it is actually true, we have committed a Type I Error When we fail to reject the null hypothesis, if it was actually false, we have committed a Type II Error Truth H 0 trueH 0 false Our decision Reject H 0 Type I error FTR H 0 Type II error

What is the probability of a Type I Error? H 0 :  = 2/3 vs. H a :  ≠ 2/3 How often will we come to the decision to reject? “Rejection Region” >.749 <.584

What is the probability of a type II error? H 0 :  = 2/3 vs. H a :  ≠ 2/3 How often will we come to the decision to fail to reject when we shouldn’t? Well first, what is the right value? What if  =.5? How often fail to reject?

Type I Error Type II Error Power

Interpretation If  actually equals.5, then there is a very high probability we will reject H 0 :  = 2/3. This sample size seems reasonably sensitive to a difference that large. But what if  actually equals.6? How often will we be able to detect that (correct reject 2/3)?

P. 319 Part (l)

For Thursday Part (l) on p. 319 PP (mostly d) HW 5 Miniproject 2 proposals next week