1. Review of Hypothesis Testing: –see Figures 7.3 & 7.4 on page 239 for an important issue in testing the hypothesis that  =20. There are two types of error that one could make (Type I and Type II – see the box at the bottom of p. 239) –the text gives 5 parts to a typical hypothesis test (and I’ll add a 6 th ): set up the null and alternative hypotheses specify the probability of a Type I error,  (and possibly  ) set up the rejection region; i.e., an interval based on the sampling distribution of the test statistic that tells us what values of the TS will lead to rejection of the null hypothesis calculate the value of the TS from the data use the value of the TS and the rejection region to make a decision about rejecting the null hypothesis or not write an interpretation of the results – what does the decision mean in the context of the problem? Read the discussion of this in section 7.4 Do # 7.27-7.29, 7.31-7.34

2. Now let’s get specific to hypothesis testing for one mean: –null hyp:  =  0 ; alternative hyp: choose from 3 possible (p.246) –rejection region (or critical region): choose from 3 (table p.246) –go over the calculations, make decision based on value of . –for review, work through the example on p.246-247 P-value approach: go through the steps in the classical approach, but evaluate the probability of getting a value “more extreme” than the one we got (assuming the null hyp. is true) – this is the p-value and generally “small” values lead to rejection of the null, “large” values cause us not to reject the null. Large sample vs. small sample testing of mu: –use Z statistic when the sample size is large and population s.d. is known –use t statistic when the sample size is small and population s.d. is unknown... work through the example on p.250-251

3. Let’s give some simple guidelines (from D. Moore) as to when the t statistic should be used... –sample size less than 15: use t procedures if the data are close to normal. If the data are clearly nonnormal or if outliers are present, don’t use t –sample size at least 15: the t procedures can be used except in the presence of outliers or strong skewness –large samples: the t procedures can be used even for clearly skewed distributions when the sample is large, roughly >=40. An important relationship exists between a 95% confidence interval for the mean and a.05 level 2-sided hypothesis test for the mean: don’t reject any null hypothesis    at the.05 level if    is in the 95% confidence interval for .

4. The Operating Characteristic (OC) Curve is a plot of L(  )=P(accepting the null hyp. when  is the true mean) against . So if  is a value of the mean for which the null is true, 1- L(  ) = P(Type I error). When  is a value of the mean for which the alternative is true, then L(  ) = P(Type II error). Thus the OC function (or the OC curve when plotted) contains all the information about both types of error... See Figures 7.9 and 7.10 for plots of the OC curve in the one-tailed and two-tailed tests respectively... study these plots very carefully and work through the examples on pages 255-256. HW: Read sections 7.5-7.7; do #7.39, 41, 43, 45, 47-52,54, 56; can you write R code to compute P(Type II errors)?