5 Basic Steps in Any Hypothesis Test

Transcription

1 5 Basic Steps in Any Hypothesis Test Step 1: Determine hypotheses (H0 and Ha). H0: μ d = 0 (μ 1 μ 2 =0) Ha: μ d > 0 (μ 1 μ 2 >0) upper-sided Ha: : μ d 0 (μ 1 μ 2 0) two-sided Step 2: Verify necessary conditions, compute an appropriate test statistic. Step 3: Assuming H0 is true, find Decision Rule Step 4: Decide whether or not Reject H0. Ha: μ d <0 (μ 1 μ 2< 0) lower-sided Step 5: Report the conclusion in the context of the problem.

2 Population Mean Difference Test Scenario Data Population Parameter Sample Statistics Response Explanatory Variable Population Mean Difference 2 Dependent Samples (on the same set of objects) Numerical (GPA, Weight) Treatment Is there a significant improvement (difference) in SAT-Math score after taking AP Calculus? Is certain weight loss program effective? Can you save buying books on Amazon compared to Borders?

3 Hypothesis Test for Population Mean Difference (H0: μ d =μ d0 ) Required assumptions: Data are 2 dependent random samples (on the same set of objects) Large n( 30) or Normal Distribution of differences X 1 - X 2 Test Statistics:

4 Rejection Rule H0: μ d = 0 (μ 1 μ 2 =0) Ha: μ d > 0 (μ 1 μ 2 >0) upper-sided Reject H0 if Z Z 1-α (t t 1-α, df ) Ha: μ d 0 (μ 1 μ 2 0) two-sided Reject H0 if Z Z 1-α/2 (t t 1-α/2,df ) or Z - Z 1-α/2 (t - t 1-α/2,df ) Ha:μ d <0 (μ 1 μ 2 <0) lower-sided Reject H0 if Z - Z 1-α (t - t 1-α,df)

5 Testing Effectiveness of a Weight Loss Program. A nutrition expert is examining a weight-loss program to evaluate its effectiveness (participants lose weight on the program). n=10 subjects are randomly selected for the investigation. The subjects' initial weights are recorded, they follow the program for 6 weeks, and they are weighted again.

6 Testing Effectiveness of a Weight Loss Program (n=10). Data: Subjects Initial Weight Final Weight Difference(d) =

7 Testing Effectiveness of a Weight Loss Program Data Summaries: Sample mean difference: Sample std of difference:

8 n= 10 people Question: is a weight-loss program effective? Step1: Parameter: H0: Ha: Significance level α =

9 n= 10 people Question: is a weight-loss program effective? Step2: Check Assumptions: Compute Test Statistic: df=n-1=

10 n= 10 people Question: is a weight-loss program effective? Step3: Reject H0 if

11 Rejection Rule H0: μ=μ 0 (p=p 0 ) Ha: μ>μ 0 (p>p 0 ) upper-sided Reject H0 if Z Z 1-α (t t 1-α, df ) Ha: μ μ 0 (p p 0 ) two-sided Reject H0 if Z Z 1-α/2 (t t 1-α/2,df ) or Z - Z 1-α/2 (t - t 1-α/2,df ) Ha: μ<μ 0 (p<p 0 ) lower-sided Reject H0 if Z - Z 1-α (t - t 1-α,df)

12 n= 10 people Question: is a weight-loss program effective? Step4: Decision: Reject H0 Fail to reject H0

13 n= 10 people Question: is a weight-loss program effective? Step5: Conclusion: Based on the sample of n=, there significant evidence, at level α=, to conclude that, on average.

14 n= 10 people Question: How effective is a weight-loss program on average? Confidence Interval (90% CI):

15 Exercise Testing Effect of Antihistamine and Alcohol on Functioning. Several years ago there were concerns about patients' appropriate use of over- the-counter medications, in particular the effects on functional abilities over-the-counter antihistamines taken (inappropriately) in combination with alcohol. An investigation of these effects was undertaken according to crossover design.

16 Exercise Testing Effect of Antihistamine and Alcohol on Functioning. In a crossover design, each participant is given each treatment under Each subject had functional ability measured in the presence of alcohol and the antihistamine also had functional ability measured in the presence of alcohol and a placebo! The order of treatments (i.e., alcohol and antihistamine, alcohol and placebo) was randomly assigned to eliminate carryover effects.

17 Testing Effect of Antihistamine and Alcohol on Functioning. n=100 of subjects were involved in the investigation. Difference: Time under the influence of alcohol and antihistamine Time under the influence of alcohol placebo.

18 Response: Time alcohol & antihistamine - Time alcohol & placebo Number of subjects n=100 Mean difference in times = 25 sec Std in difference in times = 20 sec Step1: Parameter: H0: Ha: Significance level α =

19 Response: Time alcohol & antihistamine - Time alcohol & placebo Number of subjects n=100 Mean difference in times = 25 sec Std in difference in times = 20 sec Step2: Assumptions: Test Statistic:

20 Hypothesis Test for Population Mean Difference (H0: μ d =μ d0 ) Required assumptions: Data are 2 dependent random samples (on the same set of objects) Large n( 30) or Normal Distribution of differences X 1 - X 2 Test Statistics:

21 Response: Time alcohol & antihistamine - Time alcohol & placebo Number of subjects n=100 Mean difference in times = 25 sec Std in difference in times = 20 sec Step3: Decision Rule: Reject H0 if

22 Rejection Rule H0: μ=μ 0 (p=p 0 ) Ha: μ>μ 0 (p>p 0 ) upper-sided Reject H0 if Z Z 1-α (t t 1-α, df ) Ha: μ μ 0 (p p 0 ) two-sided Reject H0 if Z Z 1-α/2 (t t 1-α/2,df ) or Z - Z 1-α/2 (t - t 1-α/2,df ) Ha: μ<μ 0 (p<p 0 ) lower-sided Reject H0 if Z - Z 1-α (t - t 1-α,df)

23 Response: Time alcohol & antihistamine - Time alcohol & placebo Number of subjects n=100 Mean difference in times = 25 sec Std in difference in times = 20 sec Step4: Decision: Reject H0 Fail to Reject H0

24 Question of Interest: Is there an effects on functional abilities over-the-counter antihistamines taken (inappropriately) in combination with alcohol? Step5: Conclusion: Based on the sample of n=, there significant evidence, at level α=, to conclude that, on average.