1 ConceptsDescriptionHypothesis TheoryLawsModel organizesurprise validate formalize The Scientific Method

2 Hypothesis Testing Population parameter = hypothesized? One sample mean = another sample mean? Null hypothesis

3 Hypothesis Testing One-sample tests –One-sample tests for the mean –One-sample tests for proportions Two-sample tests –Two-sample tests for the mean

4 Hypothesis Testing Confidence interval  Interval Hypothesis testing  Particular, predetermined value

5 Hypothesis Testing Hypothesis testing  Null hypothesis Purpose  Test the viability Null hypothesis  Population parameter  Reverse of what the experimenter believes

6 Hypothesis Testing 1. State the null hypothesis, H 0 2. State the alternative hypothesis, H A 3. Choose a, our significance level 4. Select a statistical test, and find the observed test statistic 5. Find the critical value of the test statistic 6. Compare the observed test statistic with the critical value, and decide to accept or reject H 0

7 Hypothesis Testing – Step 1 1.State the null hypothesis (H 0 ) –H 0 : μ = μ 0 –H 0 : μ - μ 0 = 0

8 Hypothesis Testing – Step 2 2. State the alternative hypothesis –H A : μ # μ 0  two-sided (two-tailed) or –H A : μ > μ 0 –H A : μ < μ 0  one-sided (one-tailed) upper-tailed lower-tailed

9 Hypothesis Testing – Step 3 3. Choose α, our significance level –It really depends on what we are testing –α = 0.05 –α = 0.01 –Type I error

10 Hypothesis Testing - Errors Type I Error - α error, occurs when we reject the null hypothesis when we should accept it Type II Error - β error, occurs when we accept the null hypothesis when we should reject it

11 Hypothesis Testing - Errors H 0 is true H 0 is false Accept H 0 Correct decisionType II Error (β) (1-α) Reject H 0 Type I Error (α)Correct decision (1-β)

12 Hypothesis Testing – Step 4 4. Select a statistical test, and find the test statistic Test statistic =  -  0 Std. error 

13 Hypothesis Testing – Step 4 4. Select a statistical test, and find the test statistic Test statistic =  -  0 Std. error 

14 Hypothesis Testing – Step 5 5. Find the critical value of the test statistic –Standard normal table –Student’s t distribution table –Two-sided vs. one-sided

15 Two-sided tests  Z α/2

16 One-sided tests  Z α

17 Hypothesis Testing – Step 6 6. Compare the observed test statistic with the critical value | Z test | > | Z crit |  H A | Z test |  | Z crit |  H 0 Z crit -Z crit H0H0 HAHA HAHA

18 | Z test | > | 1.96 |  H A | Z test |  | 1.96 |  H H0H0 HAHA HAHA Hypothesis Testing – Step 6 6. Compare the observed test statistic with the critical value

19 Hypothesis Testing – Step 6 Z test > Z crit  H A Z test  Z crit  H 0 Z crit H0H0 HAHA 6. Compare the observed test statistic with the critical value

20 Hypothesis Testing – Step 6 Z test >  H A Z test   H H0H0 HAHA 6. Compare the observed test statistic with the critical value

21 p-value p-value is the probability of getting a value of the test statistic as extreme as or more extreme than that observed by chance alone, if the null hypothesis H 0, is true. It is the probability of wrongly rejecting the null hypothesis if it is in fact true It is equal to the significance level of the test for which we would only just reject the null hypothesis

22 p-value p-value vs. significance level Small p-values  the null hypothesis is unlikely to be true The smaller it is, the more convincing is the rejection of the null hypothesis

23 One-Sample z-Tests