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### 24 tests

1. 1. Stat310 Hypothesis tests Hadley Wickham Wednesday, 21 April 2010
2. 2. 1. Quiz 2. Final 3. Hypothesis tests Wednesday, 21 April 2010
3. 3. Final 2 hour take home Due last day of exam period Larger number of smaller questions like the homeworks More info on Thursday Wednesday, 21 April 2010
4. 4. 1. Write down Ho and Ha (positions of defence and prosecution) 2. Figure out good test statistic (what numeric summary?) 3. Work out null distribution (distribution of innocents) 4. Calculate p-value by comparing actual value to null distribution (what proportion of true innocents look more guilty than the suspect) 5. Reject Ho if p-value smaller than cutoff Wednesday, 21 April 2010
5. 5. Hypothesis Null hypothesis = Ho Alternative hypothesis = Ha Hint: because we need to be able to calculate the null distribution, the null hypothesis will always be of the form: Some parameter = some value μ=0 Wednesday, 21 April 2010
6. 6. Suitcase Does this suitcase contain a radioactive bomb? Construct Ho and Ha. Let R be the background radiation measured over a minute. R ~ Poisson(2). Let S be the radiation from the suitcase. Construct a more precise Ho and Ha. Wednesday, 21 April 2010
7. 7. Grade difference I’m interested in whether or not there is a difference between this years average stat310 grade and last years. Construct Ho and Ha. If grades are normally distributed both years, can you rewrite the null hypothesis to be more precise? What other assumptions do you need? Wednesday, 21 April 2010
8. 8. Alternative Can be one-sided or two-sided Basically comes down to the question of what more guilty means. For the suitcase: guilty means higher radiation. For course grades: guilty means positive or negative difference. Wednesday, 21 April 2010
9. 9. 1. Write down Ho and Ha (positions of defence and prosecution) 2. Figure out good test statistic (what numeric summary?) 3. Work out null distribution (distribution of innocents) 4. Calculate p-value by comparing actual value to null distribution (what proportion of true innocents look more guilty than the suspect) 5. Reject Ho if p-value smaller than cutoff Wednesday, 21 April 2010
10. 10. Xi iid, and n large (> 30): ¯n − µ . X √ ∼Z σ/ n ¯n − µ . X √ ∼Z S/ n Wednesday, 21 April 2010
11. 11. iid 2 Xi ∼ Normal(µ, σ ) (n − 1)S2 2 ∼ χ (n − 1) 2 σ X ¯n − µ √ ∼Z σ/ n X ¯n − µ √ ∼ tn−1 s/ n Wednesday, 21 April 2010
12. 12. Others Difference of normals Sum of poisson Sum of binomial ... Wednesday, 21 April 2010
13. 13. Suitcase What test statistic might you use? (What experiment might you conduct?) What is the null distribution? Wednesday, 21 April 2010
14. 14. Grade difference What test statistic might you use? What is its null distribution? Wednesday, 21 April 2010
15. 15. 1. Write down Ho and Ha (positions of defence and prosecution) 2. Figure out good test statistic (what numeric summary?) 3. Work out null distribution (distribution of innocents) 4. Calculate p-value by comparing actual value to null distribution (what proportion of true innocents look more guilty than the suspect) 5. Reject Ho if p-value smaller than cutoff Wednesday, 21 April 2010
16. 16. P-value Standardised measurement of evidence. Low p-value = low probability of innocent looking this guilty = reject the null High p-value = high probability of innocent looking this guilty = don’t reject Don’t need to know anything else about the test! Wednesday, 21 April 2010
17. 17. Suitcase X ∼ Poisson(2) Background radiation is X P(X ≤ x) Poisson(2) 0 0.14 If I measure the suitcase 1 0.41 and record a 3, what’s the p 2 0.68 -value? What if I record a 5? 3 0.86 What’s the probability it’s a 4 0.95 bomb? 5 0.98 Wednesday, 21 April 2010
18. 18. Course grades Assume for simplicity there were 100 students both years, and the variance of the course grade was 80. What would the distribution of the test statistic be? Wednesday, 21 April 2010
19. 19. 1. Write down Ho and Ha (positions of defence and prosecution) 2. Figure out good test statistic (what numeric summary?) 3. Work out null distribution (distribution of innocents) 4. Calculate p-value by comparing actual value to null distribution (what proportion of true innocents look more guilty than the suspect) 5. Reject Ho if p-value smaller than cutoff Wednesday, 21 April 2010
20. 20. Say is Say is guilty innocent False Is guilty Correct acquittal False Is innocent Correct conviction Wednesday, 21 April 2010
21. 21. Your turn Which type of error is more expensive/ more costly/worse in the criminal justice system? Wednesday, 21 April 2010
22. 22. Reject HO Accept HO Type II HO false Correct error Type I HO true Correct error Wednesday, 21 April 2010
23. 23. Rates For a given test, P(false conviction) = α = signiﬁcance level P(false acquittal) = β = power What do think happens to β if you try to make α smaller? Wednesday, 21 April 2010
24. 24. α↑ β↓ α↓ β↑ Wednesday, 21 April 2010
25. 25. Cut off Choose cut-off based on rate of false convictions. If you want a 5% rate of false convictions, reject Ho if the p-value is less than 0.05. Can work out power. Wednesday, 21 April 2010
26. 26. Connection to conﬁdence intervals If you construct a 90% conﬁdence interval, and it doesn’t include the parameter until the null, then the p-value must be > 1 - 0.9 = 0.1. If the p-value is 0.08, then a 92% or greater conﬁdence interval would include the null parameter, and a smaller conﬁdence interval would not. Wednesday, 21 April 2010
27. 27. Next time Last notes on testing. More info about the ﬁnal and study sessions. Opportunity for feedback. Why statistics is awesome and you should do more. Class party! Wednesday, 21 April 2010