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 Alternative hypothesis (H 1 ): p  0.5

k p(k) 95% EXPERIMENT

Experiment P[ k 32 ] < 0.05 If k 32 then an event happened with probability < 0.5 Improbable enough to REJECT the hypothesis H 0

Test construction 1832 accept reject

k Cpdf(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)

k cpdf(k)

Significance level P[ k  10 or k  40 ]  We REJECT the hypothesis H 0 at significance level p=

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 H 0 - sample = sample from normal distribution with  = 5cm Alternative H 1 - sample = sample from normal distribution with  > 5cm

Construction of the test t t 0.95 reject 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 Are these data sampled from population with exponential pdf ?

Construction of the  2 test p1p1 p2p2 p3p3 p4p4

Construction of the test 22  reject Cannot reject

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

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-a a = significance level b 1-b = power of the test