This document introduces hypothesis testing, including: - Formulating null and alternative hypotheses for tests involving population means and proportions - Using test statistics, critical values, and p-values to test hypotheses - Defining Type I and Type II errors and their probabilities - Examples of hypothesis tests for means (using z-tests and t-tests) and proportions (using z-tests) are provided to illustrate the concepts.

1. Introduction to Hypothesis Testing

7. Population Claim: the population mean age is 50. (Null Hypothesis: REJECT Suppose the sample mean age is 20: x = 20 Sample Null Hypothesis 20 likely if  = 50?  Is Hypothesis Testing Process If not likely, Now select a random sample H 0 :  = 50 ) x

8. Reason for Rejecting H 0 Sampling Distribution of x  = 50 If H 0 is true If it is unlikely that we would get a sample mean of this value ... ... then we reject the null hypothesis that  = 50. 20 ... if in fact this were the population mean… x

10. Level of Significance and the Rejection Region H 0 : μ ≥ 3 H A : μ < 3 0 H 0 : μ ≤ 3 H A : μ > 3 H 0 : μ = 3 H A : μ ≠ 3   /2 Represents critical value Lower tail test Level of significance =  0 0  /2  Upper tail test Two tailed test Rejection region is shaded

13. Outcomes and Probabilities State of Nature Decision Do Not Reject H 0 No error (1 - )  Type II Error ( β ) Reject H 0 Type I Error ( )  Possible Hypothesis Test Outcomes H 0 False H 0 True Key: Outcome (Probability) No Error ( 1 - β )

17. Lower Tail Tests Reject H 0 Do not reject H 0 The cutoff value, or , is called a critical value  -z α x α -z α x α 0 μ H 0 : μ ≥ 3 H A : μ < 3

18. Upper Tail Tests Reject H 0 Do not reject H 0 The cutoff value, or , is called a critical value  z α x α z α x α 0 μ H 0 : μ ≤ 3 H A : μ > 3

19. Two Tailed Tests Do not reject H 0 Reject H 0 Reject H 0 There are two cutoff values ( critical values ): or  /2 -z α/2 x α/2 ± z α/2 x α/2 0 μ 0 H 0 : μ = 3 H A : μ  3 z α/2 x α/2 Lower Upper x α/2 Lower Upper  /2

21. Calculating the Test Statistic  Known Large Samples  Unknown Hypothesis Tests for μ Small Samples The test statistic is:

22. Calculating the Test Statistic  Known Large Samples  Unknown Hypothesis Tests for  Small Samples The test statistic is: But is sometimes approximated using a z: (continued)

23. Calculating the Test Statistic  Known Large Samples  Unknown Hypothesis Tests for  Small Samples The test statistic is: (The population must be approximately normal) (continued)

25. Hypothesis Testing Example Test the claim that the true mean # of TV sets in US homes is at least 3. (Assume σ = 0.8) 1. Specify the population value of interest The mean number of TVs in US homes 2. Formulate the appropriate null and alternative hypotheses H 0 : μ  3 H A : μ < 3 (This is a lower tail test) 3. Specify the desired level of significance Suppose that  = .05 is chosen for this test

36. Review: Finding Critical Value - One Tail Z .07 .09 1.1 .3790 .3810 .3830 1.2 .3980 .4015 1.3 .4147 .4162 .4177 z 0 1.28 . 08 Standard Normal Distribution Table (Portion) What is z given  = 0.10?   = .10 Critical Value = 1.28 .90 .3997 .10 .40 .50

53. Using PHStat Options

54. Sample PHStat Output Input Output