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z-test and t-test
Xuhua Xia
[email protected]
http://dambe.bio.uottawa.ca
Properties of a Normal Distribution
68.27% of the measurements lie within the range of ,
95.44% lie within 2,
99.73% lie within 3,
50% lie within 0.67,
95% lie within 1.96,
97.5% lie within 2.24,
99% lie within 2.58,
99.5% lie within 2.81,
99.9% lie within 3.29.
Given  = 70kg and  = 10kg for a normal distribution (of body weight), what is
the probability of a body weight of 40 kg belonging to the population?
The normal deviate: Z 
Xi  

Standard deviation and Standard Error of the mean:  X 
2

n
X 
The standard deviate pertaining to the normal distribution of means: Z  i
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n

X
The z-score
Z
Xi  
X
 1.96
The government has certain regulations on
commercial product. Suppose that packages of sugar
labeled as 2 kg should have a mean weight of 2 kg
and a standard deviation equal to 0.10. If a package
of sugar labeled 2 kg that you bought from a store
has a weight of 1.82 kg, what is the z score? Can
you present the package as evidence that the
manufacturer has violated the government
regulation?
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Normal Distribution
Body Weight
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106.89
100.47
94.06
87.64
81.23
74.81
68.40
61.98
55.57
49.15
42.74
36.32
350
300
250
200
150
100
50
0
29.91
Frequency
Body Weight of 10,000 Adult Men
Mean = 70 kg, Std Dev = 10 kg
350
300
250
200
150
100
50
0
Body Weight
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106.89
100.47
94.06
87.64
81.23
74.81
68.40
61.98
55.57
49.15
42.74
36.32
s
sx 
n
29.91
Frequency
Frequency Distribution of Means
Darwin’s Breeding Experiment
Wrong method assuming
normal distribution:
 = 20.933;  = 37.744; n = 15;
X 
Z

n

Xi  
X
37.744
 9.75
15

20.933
 2.147  1.96
9.75
Therefore, the mean difference
is significantly larger than
zero, i.e., inbreeding does
reduce seed production.
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Species Outbreed Inbreed Difference
100
51
49
1
222
289
-67
2
121
113
8
3
433
417
16
4
222
216
6
5
111
88
23
6
534
506
28
7
432
391
41
8
99
85
14
9
445
416
29
10
112
56
56
11
333
309
24
12
222
147
75
13
422
362
60
14
101
149
-48
15
Is the mean difference
significantly larger than 0?
Problem of Small Samples
I may premise that if we took by
chance a dozen or score of men
belonging to two nations and
measured them, it would I presume
be very rash to form any judgment
from such small numbers on their
(the nation’s) average heights. But
the case is somewhat different with
my … plants, as they were exactly of
the same age, were subjected from
first to last to the same conditions,
and were descended from the same
parents.
-- Darwin, quoted in Fisher’s The design of experiments.
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Species Outbreed Intbreed Difference
100
51
49
1
222
289
-67
2
121
113
8
3
433
417
16
4
222
216
6
5
111
88
23
6
534
506
28
7
432
391
41
8
99
85
14
9
445
416
29
10
112
56
56
11
333
309
24
12
222
147
75
13
422
362
60
14
101
149
-48
15
William S. Gosset & t Distribution
t distribution is wider and flatter than the normal distribution
350
300
250
200
150
100
50
0
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Body Weight
106.89
100.47
94.06
87.64
81.23
74.81
68.40
61.98
55.57
49.15
42.74
36.32
t distribution
29.91
Frequency
Normal distribution
t distribution
• The t distribution depends on the degree of freedom (DF). For
Darwin’s data with a sample size = 15, DF = 15 - 1 = 14.
• With the t distribution with DF = 14, we expect 95% of the
observations should fall within the range of mean  2.145
STD.
• Remember that for a normal distribution, 95% of the
observations are expected to fall within the range of   1.96
.
• For pair-sample t-test with the null hypothesis being Mean1 =
Mean2 (or MeanD = 0):
t
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D  0 20.933

 2.147  2.145
sX
9.75
T-Test
• T-Test can be used to test
– the difference in mean between two samples (paired or unpaired),
– a sample mean against a mean of a known population (e.g., the
concentration of a medicine set as a standard by the government),
– whether a single individual observation belong to a sample with
sample size larger than one.
• The normal distribution and the Student’s t distribution. Why
should the statistic t take into consideration both the mean
difference and the variance?
• How to apply the test using Excel or SAS.
• The assumptions.
• Alternative methods: Wilcoxon rank-sum test or MannWhitney U test.
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The Essence of the t Statistic
Same variance,
smaller mean difference
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
Same mean difference,
larger variance
-18
-12
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-6
0
6
12
18
X1  X 2
t
pooled SX
6
More on variance and SE
Two independent variables: x1, x2 sampled from two normal distributions
sx21  x2  E ( sx21  sx22 )
sx21  x2  E ( sx21  sx22 )
A better estimate:
s
2
x1  x2
 SS1 SS2  SS1  SS2
 E (s  s )  E 


DF
DF
DF1  DF2
 1
2 
S x1 
2
x1
sx21
n1
; S x2 
2
x2
sx22
n2
with n1  n2  n : S x1  x2 
sx21  sx22
n
with n1  n2 , but both large: S x1  x2 
sx21
n1

Estimate of S x1  x2 assuming equal variance:
S x1  x2 
sx21  x2
n1
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
sx21  x2
n2
sx22
n2
Computation for unpaired t-test
Sample 1
Sample 2
Sample size
n1
n2
Mean
x1
x2
Standard dev.
s1
s2
Sample size
7
7
Mean
76.857
82.714
Standard dev.
2.545
3.147
t
(76.857  82.714)
2.545  3.147
7
2
2
 3.828
Df = (7-1) + (7-1) = 12
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x1  x2
t
S x1  x2
with n1  n2  n : S x1  x2 
sx21  sx22
n
with n1  n2 , but both large: S x1  x2 
sx21
n1

Estimate of S x1  x2 assuming equal variance:
S x1  x2 
sx21  x2
n1

sx21  x2
n2
sx22
n2
Paired-sample t-test: 3
Using blocks to reduce confounding environmental factors (Everything else
being equal except for the treatment effect) in evaluating the protein content
of two wheat variaties.
1
1
1
1
1
2
1
2
1
2
1
2
2
2
2
2
Block 1
Block 2
Block 3
Block 4
1
2
2
1
2
2
1
1
1
2
2
1
2
1
2
1
Block 1
Block 2
Block 3
Block 4
How should we allocate the two crop varieties to the plots? What comparison would be fair?
Xuhua Xia
The Wilcoxon-Mann-Whitney Test
• Statistical significance tests can be grouped into
– Parametric tests, e.g., t-test, ANOVA
– Non-parametric tests, e.g., Wilcoxon-Mann-Whitney test,
sign test, runs test.
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When to Use Non-parametric Tests
• Parametric tests depends on the assumed probability
distributions, e.g., normal distribution, t distribution,
etc, and would give misleading results when the
assumptions are violated.
• Non-parametric tests are called distribution-free tests
and can be used in cases where the parametric tests
are inappropriate.
• Parametric tests are more powerful than their nonparametric counterparts when the underlying
assumptions are met.
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Wilcoxon-Mann-Whitney Test
• The Wilcoxon-Mann-Whitney test is the nonparametric equivalent of the t-test.
• The original data are rank-transformed before
applying the test
• The test statistic is U
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