1 When we free ourselves of desire, we will know serenity and freedom.

2 Inferences about Population Means Sections Estimation Statistical tests: z test and t test Sample size selection

3 Estimation To estimate a numerical summary in population (parameter): Point estimator — the same numerical summary in sample; a statistic Interval estimator — a “random” interval which includes the parameter most of time

4 Confidence Interval Ideally, a short interval with high confidence interval is preferred.

5

6 Estimation for  Point estimator Confidence interval – Normality with known  or large sample:  interval – Normality with unknown  t interval

7 One Population

8 Sample Size for Estimating  Where E is the largest tolerable error. if  is unknown, use s from prior data or the upper bound of 

9 The Logic of Hypothesis Test “Assume Ho is a possible truth until proven false” Analogical to “Presumed innocent until proven guilty” The logic of the US judicial system

10 Steps in Hypothesis Test 1. Set up the null (H 0 ) and alternative (H 1 ) hypotheses 2. Find an appropriate test statistic (T.S.) 3. Find the rejection region (R.R.) 4. Reject Ho if the observed test statistic falls in R.R. 5. Report the result in the context of the situation

11 Determine Ho and H 1 The hypothesis we favor (called the research hypothesis) goes to H 1, if possible. Eg. Example 5.7 (p.238)

12 Good! Good!(Correct!) H 0 trueH 1 true Type II Error, or “  Error” Type I Error, or “  Error” Good! Good!(Correct) we accept H 0 we reject H 0 Types of Errors

13 Z Test For normal populations or large samples (n > 30) And the computed value of Z is denoted by Z*.

14 Types of Tests

15 Types of Tests

16 Types of Tests

Power of Test Example 5.7 revisit (p. 240) 17

18 Sample Size for Testing  The type I, II error rates are controlled at  respectively and the maximum tolerable error is  One-tailed tests: Two-tailed tests:

19 P-value (Level of Significance) p-value is the probability of seeing what we observe as far as (or further) from Ho (in the direction of H 1 ) given Ho is true; the smallest  level to reject Ho p-value is computed by assuming Ho is true and then determining the probability of a result as extreme (or more extreme) as the observed test statistic in the direction of the H 1. The smaller p-value is, the less likely that what we observe will occur under the assumption Ho is true. Smaller p-value means stronger evidence against Ho.

20 Computing the p-Value for the Z-Test

21 Computing the p-Value for the Z-Test

22 Computing the p-Value for the Z-Test P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|)

23 Hypothesis Test using p-Value 1. Set up the null (Ho) and alternative (H 1 ) hypotheses 2. Find an appropriate test statistic (T.S.) 3. Find the p-value 4. Reject Ho if the p-value  5. Report the result in the context of the situation

24 Example 5.7 (Page 238) Redo it using the p-value way

25 t Test For normal populations with unknown  t = the same formula for Z but replacing  by s Eg. Revisit Example 5.7

26 One Population