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## Hypotheses Testing

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**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 (H0): p = 0.5 Alternative hypothesis (H1): p 0.5**EXPERIMENT**p(k) 95% k**Experiment**P[ k < 18 or k > 32 ] < 0.05 If k < 18 or k > 32 then an event happened with probability < 0.5 Improbable enough toREJECTthe hypothesis H0**Test construction**18 32 accept reject reject**0.975**Cpdf(k) 0.025 k**Conclusion**No premise to reject the hypothesis**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)**cpdf(k)**k**Significance level**P[ k 10 or k 40 ] 0.000025 We REJECT the hypothesis H0 at significance level p=0.000025**Remark**In STATISTICS To prove something = REJECT the hypothesis that converse is true**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 X everyday, from the birth, for the period of 6 months.**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**Construction of the test**reject t t0.95 Cannot reject**We do not population variance, and/or we suspect that**vitamin treatment may change the variance – so we use t distribution**2 test (K. Pearson, 1900)**To test the hypothesis that a given data actually come from a population with the proposed distribution**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 ?**Construction of the 2 test**p2 p3 p1 p4**Construction of the test**reject 2 2 0.95 Cannot reject**How about**Are these data sampled from population with exponential pdf ? • Estimate a • Use 2 test • Remember d.f. = K-2**Power and significance of the test**Actual situation decision probability 1-a accept H0 true Reject = error t. I a = significance level b Accept = error t. II H0 false reject 1-b = power of the test