Download Biochemistry and Physiology

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
EXPERIMENT
0.12
0.1
0.08
p(k)
0.06
0.04
95%
0.02
k
0
0
5
10
15
20
25
30
35
40
45
50
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
reject
32
accept
reject
1
0.975
0.9
0.8
0.7
Cpdf(k) 0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
k
30
35
40
45
50
0.025
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)
1
0.9
0.8
0.7
0.6
cpdf(k)
0.5
0.4
0.3
0.2
k
0.1
0
0
5
10
15
20
25
30
35
40
45
50
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
1 N
X   Xi
N i 1
S
1 N
2
 X i  X 
N i 1
X 
t
N 1
S
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
0.3564
0.4048
0.1897
0.2360
1.2595
3.4827
1.3283
0.6535
0.6375
1.1692
0.0046
0.6874
1.2521
2.3923
0.6592
1.0536
0.3991
0.7658
0.4263
0.8325
0.2674
1.1440
0.5301
0.7744
0.7029
0.5572
0.6569
0.3698
0.3049
2.2836
1.4987
0.0907
2.4005
0.8774
0.1954
0.9500
1.2336
0.0552
0.7944
1.9015
0.8007
0.3137
1.0383
2.0369
0.6698
0.3075
0.1074
0.3527
0.3046
0.4425
2.6742
0.3678
0.2862
1.0939
0.3560
1.1900
0.6193
3.3593
0.9115
1.2388
0.6363
0.3923
0.2654
0.2545
0.1155
1.3249
0.4360
0.4527
0.2112
0.0326
0.1402
2.5008
0.3974
0.2938
0.5899
1.1676
0.1358
0.2192
0.1843
0.0237
0.2555
0.3712
2.8841
3.3202
1.9808
0.4713
0.1737
1.3994
0.5082
2.2617
0.0080
0.7095
1.6093
0.9300
3.2906
0.6311
1.6893
0.0769
1.4138
Are these data sampled from population with exponential pdf ?
f ( x)  e
x
Construction of the 2 test
1
0.9
p2
p1
0.8
p3
p4
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Construction of the test
reject
2
2 0.95
Cannot reject
How about
Are these data sampled from population with exponential pdf ?
f ( x)  ae
 ax
1. Estimate a
2. Use 2 test
3. Remember d.f. = K-2
Power and significance of the test
Actual
situation
decision
accept
probability
1-a
H0 true
Reject = error t. I
Accept = error t. II
a = significance level
b
H0 false
reject
1-b = power of the test