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Hypotheses Testing

Hypotheses Testing. 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). Null versus Alternative. Null hypothesis (H 0 ): p = 0.5

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Hypotheses Testing

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  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 (H0): p = 0.5 Alternative hypothesis (H1): p  0.5

  4. EXPERIMENT p(k) 95% k

  5. Significance level α= Probability of Type 1 error=Pr[rejecting H0 | H0 true] P[ k < 18 or k > 32 ] < 0.05 If k < 18 or 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 18 32 accept reject reject

  7. 0.975 Cpdf(k) 0.025 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. cpdf(k) k

  11. p-value P[ k  10 or k  40 ]  0.000025 We could REJECT hypothesis H0 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 H0 - sample = sample from normal distribution with  = 5cm Alternative H1 - sample = sample from normal distribution with  > 5cm

  15. Construction of the test reject t t0.95 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 p2 p3 p1 p4

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

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

  22. Power and significance of the test Actual situation decision probability 1-α accept H0 true Reject = error t. I α = significance level β Accept = error t. II H0 false reject 1-β = power of the test

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