Hypotheses Testing

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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)

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Null versus Alternative

• Null hypothesis (H0) p 0.5
• Alternative hypothesis (H1) p ? 0.5

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EXPERIMENT
p(k)
95
k
5
Experiment

• P k lt 18 or k gt 32 lt
  0.05
• If k lt 18 or k gt 32 then an
  event
• happened with probability lt 0.5
• Improbable enough to REJECT the hypothesis H0

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Test construction
18
32
accept
reject
reject
7
0.975
Cpdf(k)
0.025
k
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Conclusion

• No premise to reject the hypothesis

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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)

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cpdf(k)
k
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Significance level

• P k ? 10 or k ? 40 ? 0.000025
• We REJECT the hypothesis H0
• at significance level p 0.000025

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Remark

• In STATISTICS
• To prove something REJECT the hypothesis that
  converse is true

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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.

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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 ? gt 5cm

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Construction of the test
reject
t
t0.95
Cannot reject
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We do not population variance, and/or we suspect
that vitamin treatment may change the variance
so we use t distribution
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?2 test (K. Pearson, 1900)

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

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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 ?
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Construction of the ?2 test
p2
p3
p1
p4
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Construction of the test
reject
?2
?2 0.95
Cannot reject
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How about
Are these data sampled from population with
exponential pdf ?

1. Estimate a
2. Use ?2 test
3. Remember d.f. K-2

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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