# 1 more on Hypothesis Testing. 2 Hypothesis Testing Hypothesis Testing Trial by jury.

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1 more on Hypothesis Testing

2 Hypothesis Testing Hypothesis Testing Trial by jury

3 Individual on trial. Is he/she innocent? Evidence Trial Hypothesis Testing & Trial by jury

4 Individual on trial. Is he/she innocent? Evidence Trial Person InnocentGuilty Hypothesis Testing & Trial by jury

5 Individual on trial. Is he/she innocent? Evidence Trial Jury Person InnocentGuilty Not Guilty Guilty Hypothesis Testing & Trial by jury

6 Individual on trial. Is he/she innocent? Evidence Trial Jury Person InnocentGuilty Not Guilty  Guilty  Hypothesis Testing & Trial by jury

7 Individual on trial. Is he/she innocent? Evidence Trial Jury Person InnocentGuilty Not Guilty  x Guiltyx  Hypothesis Testing & Trial by jury

8 Evidence Trial Jury Person InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? EvidenceTrial Hypothesis Testing

9 Trial Jury Person InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? SampleTrial Hypothesis Testing

10 Jury Person InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

11 Jury Population InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

12 Jury Population  =  0 Guilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

13 Jury Population  =  0   0  0 Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

14 Us Population  =  0   0  0 Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

15 Us Population  =  0   0  0 Not reject  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

16 Us Population  =  0   0  0 Not reject  x Rejectx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

17 Us Population  =  0   0  0 Not reject  Type II RejectType I  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

18 Probability of Type I error is  i.e. the probability of rejecting the null hypothesis when it is true. Probability of Type II error is  i.e the probability of not rejecting the null hypothesis when it is false. 1-  is the power of the test. Possible errors in analysis results

19 1o1o Hypothesize a value (  0 ) 2o2o Take a random sample (n). 3o3o Is it likely that the sample came from a population with mean  0 (  = 0.05) ? Hypothesis testing about  :

20 We know that cholesterol levels in US men 20-74 yrs are normally distributed with σ X  46 mg/100ml and μ = 211. We obtain a random sample of 12 hypertensive smokers and obtain a sample mean of 217 mg/100ml. We want to test whether their population mean is the same as that of the general population? 2 sided hypothesis test - Illustration H 0 :  = 211 H A :   211

21 H 0 :  = 211 H A :   211  = 46 mg/100ml 12 hypertensive smokers have: 2 sided hypothesis test - Illustration

22 Some prefer to quote the p-value. The p-value answers the question, “What is the probability of get- ting as large, or larger, a Discrepancy given the null hypothesis is true?” P-value Question: Do hypertensive smokers have the same mean as the general population?

23 Rejecting the null hypothesis Assume a specific threshold of Type I error, α –Typically α = 0.05 If p value < α  Reject null

24 Some prefer to quote the p-value. The p-value answers the question, “What is the probability of get- ting as large, or larger, a Discrepancy given the null hypothesis is true?” P-value Answer: Do not reject the null hypothesis. No evidence that hypertensive smokers have a different mean than general population

25 Decide on statistic: Determine which values of consonant with the hypothesis that  =  0 and which ones are not. are Look atand decide. Summary

26 Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities: 1. Two-sided H 0 :  =  0 H A :    0 Alternative hypothesis

27 Blood glucose level of healthy persons has  = 9.7 mmol/L and  = 2.0 mmol/L H 0 :   9.7 H A :  > 9.7 Sample of 64 diabetics yields Example - One-sided alternative Do diabetics have blood glucose levels that are higher on average when compared to the general population?

28 Blood glucose level of healthy persons has  = 9.7 mmol/L and  = 2.0 mmol/L H 0 :   9.7 H A :  > 9.7 n = 64 p-value << 0.001 Example - One-sided alternative Answer: Reject the null hypothesis. Significant evidence that diabetics have a higher mean level of glucose when compared to the general population

29 Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities: Two-sided H 0 :  =  0 H A :    0 One-sided H 0 :    0 H A :  <  0 One-sided H 0 :    0 H A :  >  0 Alternative hypothesis

30 Summary Hypothesis testing: –Type I and II errors –Power Two sided hypothesis test One sided hypothesis test

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