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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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example 1
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 versus Alternative

Null hypothesis (H0): p = 0.5

Alternative hypothesis (H1): p  0.5

slide4

EXPERIMENT

p(k)

95%

k

experiment
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
Test construction

18

32

accept

reject

reject

slide7

0.975

Cpdf(k)

0.025

k

conclusion
Conclusion

No premise to reject the hypothesis

example 2
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)

significance level
Significance level

P[ k  10 or k  40 ]  0.000025

We REJECT the hypothesis H0

at significance level p=0.000025

remark
Remark

In STATISTICS

To prove something = REJECT the hypothesis that converse is true

example 3
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 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
Construction of the test

reject

t

t0.95

Cannot reject

slide16
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
2 test (K. Pearson, 1900)

To test the hypothesis that a given data actually come from a population with the proposed distribution

slide18
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 test20
Construction of the test

reject

2

2 0.95

Cannot reject

how about
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
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