1. Hypotheses Testing

2. Example 1 We have tossed a coin 50 times and we got k = 19 heads Should we accept/reject the hypothesis that p = 0.5 (the coin is fair)

3. Null versus Alternative Null hypothesis (H 0 ): p = 0.5 Alternative hypothesis (H 1 ): p  0.5

4. k p(k) 95% EXPERIMENT

5. Significance level α = Probability of Type 1 error =Pr[rejecting H 0 | H 0 true] P[ k 32 ] < 0.05 If k 32 then under the null hypothesis the observed event falls into the rejection region with probability α < 0.05 We want α as small as possible

6. Test construction 1832 accept reject

7. 0.025 0.975 k Cpdf(k)

8. Conclusion No evidence to reject the null hypothesis

9. Example 2 We have tossed a coin 50 times and we got k = 10 heads Should we accept/reject the hypothesis that p = 0.5 (the coin is fair)

10. k cpdf(k)

11. p-value P[ k  10 or k  40 ]  0.000025 We could REJECT hypothesis H 0 at significance level as low as α= 0.000025 p-value is the lowest attainable sig level

12. Remark In STATISTICS To prove something = REJECT the hypothesis that converse is true

13. Example 3 We know that on average mouse tail is 5 cm long. We have a group of 10 mice, and give to each of them a dose of vitamin T everyday, from the birth, for the period of 6 months.

14. We want to prove that vitamin X makes mouse tail longer We measure tail lengths of our group and we get sample = 5.5, 5.6, 4.3, 5.1, 5.2, 6.1, 5.0, 5.2, 5.8, 4.1 Hypothesis H 0 - sample = sample from normal distribution with  = 5cm Alternative H 1 - sample = sample from normal distribution with  > 5cm

15. Construction of the test t t 0.95 reject Cannot reject

16. We do not know population variance, and/or we suspect that vitamin treatment may change the variance – so we use t distribution

17.  2 test (K. Pearson, 1900) To test the hypothesis that a given data actually come from a population with the proposed distribution

18. Data 0.4319 0.6874 0.5301 0.8774 0.6698 1.1900 0.4360 0.2192 0.5082 0.3564 1.2521 0.7744 0.1954 0.3075 0.6193 0.4527 0.1843 2.2617 0.4048 2.3923 0.7029 0.9500 0.1074 3.3593 0.2112 0.0237 0.0080 0.1897 0.6592 0.5572 1.2336 0.3527 0.9115 0.0326 0.2555 0.7095 0.2360 1.0536 0.6569 0.0552 0.3046 1.2388 0.1402 0.3712 1.6093 1.2595 0.3991 0.3698 0.7944 0.4425 0.6363 2.5008 2.8841 0.9300 3.4827 0.7658 0.3049 1.9015 2.6742 0.3923 0.3974 3.3202 3.2906 1.3283 0.4263 2.2836 0.8007 0.3678 0.2654 0.2938 1.9808 0.6311 0.6535 0.8325 1.4987 0.3137 0.2862 0.2545 0.5899 0.4713 1.6893 0.6375 0.2674 0.0907 1.0383 1.0939 0.1155 1.1676 0.1737 0.0769 1.1692 1.1440 2.4005 2.0369 0.3560 1.3249 0.1358 1.3994 1.4138 0.0046 Are these data sampled from population with exponential pdf ?

19. Construction of the  2 test p1p1 p2p2 p3p3 p4p4

20. Construction of the test 22  2 0.95 reject Cannot reject

21. How about Are these data sampled from population with exponential pdf ? 1.Estimate a 2.Use  2 test 3.Remember d.f. = K-2

22. Power and significance of the test Actual situation decisionprobability H 0 true H 0 false accept Reject = error t. I reject Accept = error t. II 1-α α = significance level β 1-β = power of the test