Hypotheses Testing
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Transcript of 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 (H0): p = 0.5
Alternative hypothesis (H1): p 0.5
0 5 10 15 20 25 30 35 40 45 500
0.02
0.04
0.06
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0.1
0.12
k
p(k)
95%
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 to REJECT the hypothesis H0
Test construction
18 32
acceptreject reject
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0.025
0.975
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)
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
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k
cpdf(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
tt0.95
reject
Cannot reject
We do not population variance, and/or we suspect that vitamin treatment may
change the variance – so we use t distribution
N
iiXN
X1
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N
ii XX
NS
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21
1
NS
Xt
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 ?
xexf )(
Construction of the 2 test
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p1p2 p3 p4
Construction of the test
2
2 0.95
reject
Cannot reject
How aboutAre these data sampled from population with exponential pdf ?
axaexf )(
1. Estimate a2. Use 2 test3. Remember d.f. = K-2
Power and significance of the test
Actual situation
decision probability
H0 true
H0 false
accept
Reject = error t. I
reject
Accept = error t. II
1-a
a = significance level
b
1-b = power of the test