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1 more on Hypothesis Testing

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2 Hypothesis Testing Hypothesis Testing Trial by jury

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

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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 :

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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

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

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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?

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

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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

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

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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

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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?

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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

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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

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

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