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Probability and Statistics 2
Lecture Notes 2
Tuğba EFENDİGİL, Ph.D.
[email protected]
Agenda

Sampling distributions




Sampling distributions of Means and the Central Limit Theorem
Sampling distributions of the Difference between Two Means
Sampling distributions of a Proportion
Sampling distributions of S2
 When σ known (chi-squared distribution)
 When σ unknown (t distribution)

Sampling distributions of two sample variances (F distribution)
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions

Statistical methods are used to make decisions and draw
conclusions about populations. This aspect of statistics is
generally called statistical inference. These techniques utilize
the information in a sample in drawing conclusions.

Statistical inference may be divided into two major areas:
parameter estimation and hypothesis testing.
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions

Statistical inference is always focused on drawing conclusions
about one or more parameters of a population. An important
part of this process is obtaining estimates of the parameters.

Suppose that we want to obtain a point estimate (a reasonable
value) of a population parameter.

We know that before the data are collected, the observations
are considered to be random variables, say,

Therefore, any function of the observation, or any statistic, is
also a random variable.
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions

Since a statistic is a random variable, it has a probability
distribution. We call the probability distribution of a statistic a
sampling distribution.

The sampling distribution of a statistic depends on the




distribution of the population,
the size of the samples, and
the method of choosing the samples.
The probability distribution of
distribution of the mean.
5
is called the sampling
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of Mean and
the Central Limit Theorem [***]
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of Mean and the
Central Limit Theorem
lim f ( X ) ~ N ( ; 2 n)
n
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of Mean and the
Central Limit Theorem
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of Mean and the
Central Limit Theorem
Ex.: An electrical firm manufactures light bulbs that have a length of life that is
approximately normally distributed, with mean equal to 800 hours and a standard
deviation of 40 hours. Find the probability that a random sample of 16 bulbs will
have an average life of less than 775 hours.
Solution:
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of Mean and the
Central Limit Theorem
Ex.: Traveling between two campuses of a university in a city via shuttle bus takes,
on average, 28 minutes with a standard deviation of 5 minutes. In a given week, a
bus transported passengers 40 times. What is the probability that the average
transport time was more than 30 minutes? Assume the mean time is measured to the
nearest minute.
Solution:
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of the Difference
between Two Means
A scientist or engineer may be interested in a comparative experiment in which two
manufacturing methods, 1 and 2, are to be compared. The basis for that comparison is
μ1 − μ2, the difference in the population means.
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of the Difference
between Two Means
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of the Difference
between Two Means
Ex.: Two independent experiments are run in which two different types of paint are
compared. Eighteen specimens are painted using type A, and the drying time, in hours,
is recorded for each. The same is done with type B. The population standard deviations
are both known to be 1.0. Assuming that the mean drying time is equal for the two
types of paint, find P( 𝑋𝐴 − 𝑋𝐵 > 1.0), where 𝑋𝐴 and 𝑋𝐵 are average drying times for
samples of size nA = nB = 18.
Solution:
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of the Difference
between Two Means
Solution cont’d:
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of the Difference
between Two Means
Ex.: The television picture tubes of manufacturer A have a mean lifetime of
6.5 years and a standard deviation of 0.9 year, while those of manufacturer B
have a mean lifetime of 6.0 years and a standard deviation of 0.8 year. What
is the probability that a random sample of 36 tubes from manufacturer A will
have a mean lifetime that is at least 1 year more than the mean lifetime of a
sample of 49 tubes from manufacturer B?
Solution:
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of the Difference
between Two Means
Solution cont’d:
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of a Proportion
The estimator of a population proportion of successes is the sample proportion. That
is, we count the number of successes in a sample and compute:
X is the number of successes, n is the sample size. Note that n and p are the
parameters of a binomial distribution.
We know that the sampling distribution is approximately normal with mean p and
variance if p(1 –p)/n is not too close to either 0 or 1 and if n is relatively large.
Typically, to apply this approximation we require that np and n(1 -p) be greater than
or equal to 5.
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of a Proportion
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Known: Chi-squared distribution
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Known: Chi-squared distribution
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Known: Chi-squared distribution
Probability density functions of several chisquared distributions
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Known: Chi-squared distribution
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Lecture #2------Tuğba Efendigil, Ph.D.
Chi-squared Table
(v)
0,99
0,98
0,95
0,90
0,75
0,50
0,10
0,05
0,03
0,01
1
0,00
0,00
0,00
0,02
0,10
0,45
2,71
3,84
5,02
6,63
2
0,02
0,05
0,10
0,21
0,58
1,39
4,61
5,99
7,38
9,21
3
0,11
0,22
0,35
0,58
1,21
2,37
6,25
7,81
9,35
11,34
4
0,30
0,48
0,71
1,06
1,92
3,36
7,78
9,49
11,14
13,28
5
0,55
0,83
1,15
1,61
2,67
4,35
9,24
11,07
12,83
15,09
6
0,87
1,24
1,64
2,20
3,45
5,35
10,64
12,59
14,45
16,81
7
1,24
1,69
2,17
2,83
4,25
6,35
12,02
14,07
16,01
18,48
8
1,65
2,18
2,73
3,49
5,07
7,34
13,36
15,51
17,53
20,09
9
2,09
2,70
3,33
4,17
5,90
8,34
14,68
16,92
19,02
21,67
10
2,56
3,25
3,94
4,87
6,74
9,34
15,99
18,31
20,48
23,21
11
3,05
3,82
4,57
5,58
7,58
10,34
17,28
19,68
21,92
24,72
12
3,57
4,40
5,23
6,30
8,44
11,34
18,55
21,03
23,34
26,22
13
4,11
5,01
5,89
7,04
9,30
12,34
19,81
22,36
24,74
27,69
14
4,66
5,63
6,57
7,79
10,17
13,34
21,06
23,68
26,12
29,14
15
5,23
6,26
7,26
8,55
11,04
14,34
22,31
25,00
27,49
30,58
24
Lecture #2------Tuğba Efendigil, Ph.D.
Chi-squared Table
(v)
0,99
0,98
0,95
0,90
0,75
0,50
0,10
0,05
0,03
0,01
16
5,81
6,91
7,96
9,31
11,91
15,34
23,54
26,30
28,85
32,00
17
6,41
7,56
8,67
10,09
12,79
16,34
24,77
27,59
30,19
33,41
18
7,01
8,23
9,39
10,86
13,68
17,34
25,99
28,87
31,53
34,81
19
7,63
8,91
10,12
11,65
14,56
18,34
27,20
30,14
32,85
36,19
20
8,26
9,59
10,85
12,44
15,45
19,34
28,41
31,41
34,17
37,57
21
8,90
10,28
11,59
13,24
16,34
20,34
29,62
32,67
35,48
38,93
22
9,54
10,98
12,34
14,04
17,24
21,34
30,81
33,92
36,78
40,29
23
10,20
11,69
13,09
14,85
18,14
22,34
32,01
35,17
38,08
41,64
24
10,86
12,40
13,85
15,66
19,04
23,34
33,20
36,42
39,36
42,98
25
11,52
13,12
14,61
16,47
19,94
24,34
34,38
37,65
40,65
44,31
26
12,20
13,84
15,38
17,29
20,84
25,34
35,56
38,89
41,92
45,64
27
12,88
14,57
16,15
18,11
21,75
26,34
36,74
40,11
43,19
46,96
28
13,56
15,31
16,93
18,94
22,66
27,34
37,92
41,34
44,46
48,28
29
14,26
16,05
17,71
19,77
23,57
28,34
39,09
42,56
45,72
49,59
30
14,95
16,79
18,49
20,60
24,48
29,34
40,26
43,77
46,98
50,89
40
22,16
24,43
26,51
29,05
33,66
39,34
51,81
55,76
59,34
63,69
50
29,71
32,36
34,76
37,69
42,94
49,33
63,17
67,50
71,42
76,15
37,48
40,48
43,19
46,46
52,29
59,33
74,40
79,08
83,30
88,38
60
25
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
Chi-squared values based on probability values
Olasılıklar (1-α)
Serb.
Derecesi
0,01
0,025
0,05
0,1
0,2
0,9
0,95
0,975
0,99
1
0,00
0,00
0,00
0,02
0,06
2,71
3,84
5,02
6,63
2
0,02
0,05
0,10
0,21
0,45
4,61
5,99
7,38
9,21
3
0,11
0,22
0,35
0,58
1,01
6,25
7,81
9,35
11,34
4
0,30
0,48
0,71
1,06
1,65
7,78
9,49
11,14
13,28
5
0,55
0,83
1,15
1,61
2,34
9,24
11,07
12,83
15,09
6
0,87
1,24
1,64
2,20
3,07
10,64
12,59
14,45
16,81
7
1,24
1,69
2,17
2,83
3,82
12,02
14,07
16,01
18,48
8
1,65
2,18
2,73
3,49
4,59
13,36
15,51
17,53
20,09
9
2,09
2,70
3,33
4,17
5,38
14,68
16,92
19,02
21,67
10
2,56
3,25
3,94
4,87
6,18
15,99
18,31
20,48
23,21
11
3,05
3,82
4,57
5,58
6,99
17,28
19,68
21,92
24,72
12
3,57
4,40
5,23
6,30
7,81
18,55
21,03
23,34
26,22
13
4,11
5,01
5,89
7,04
8,63
19,81
22,36
24,74
27,69
26
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
Chi-squared values based on probability values
Probabilities (1-α)
v
0,01
0,025
0,05
0,1
0,2
0,9
0,95
0,975
0,99
14
4,66
5,63
6,57
7,79
9,47
21,06
23,68
26,12
29,14
15
5,23
6,26
7,26
8,55
10,31
22,31
25,00
27,49
30,58
16
5,81
6,91
7,96
9,31
11,15
23,54
26,30
28,85
32,00
17
6,41
7,56
8,67
10,09
12,00
24,77
27,59
30,19
33,41
18
7,01
8,23
9,39
10,86
12,86
25,99
28,87
31,53
34,81
19
7,63
8,91
10,12
11,65
13,72
27,20
30,14
32,85
36,19
20
8,26
9,59
10,85
12,44
14,58
28,41
31,41
34,17
37,57
21
8,90
10,28
11,59
13,24
15,44
29,62
32,67
35,48
38,93
22
9,54
10,98
12,34
14,04
16,31
30,81
33,92
36,78
40,29
23
10,20
11,69
13,09
14,85
17,19
32,01
35,17
38,08
41,64
24
10,86
12,40
13,85
15,66
18,06
33,20
36,42
39,36
42,98
25
11,52
13,12
14,61
16,47
18,94
34,38
37,65
40,65
44,31
26
12,20
13,84
15,38
17,29
19,82
35,56
38,89
41,92
45,64
27
12,88
14,57
16,15
18,11
20,70
36,74
40,11
43,19
46,96
28
13,56
15,31
16,93
18,94
21,59
37,92
41,34
44,46
48,28
29
14,26
16,05
17,71
19,77
22,48
39,09
42,56
45,72
49,59
27
30
14,95
16,79
Lecture
Efendigil,
18,49 #2------Tuğba
20,60
23,36 Ph.D.40,26
43,77
46,98
50,89
Sampling distributions of S2
When σ is Known: Chi-squared distribution
Ex.:
Solution:
28
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Unknown: t distribution
In many experimental scenarios, knowledge of σ is certainly no more
reasonable than knowledge of the population mean μ. Often, in fact, an
estimate of σ must be supplied by the same sample information that produced
the sample average . As a result, a natural statistic to consider to deal with
inferences on μ is
since S is the sample analog to σ.
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Unknown: t distribution
If the sample size is large enough, say n ≥ 30, the distribution of T does not
differ considerably from the standard normal.
However, for n < 30, it is useful to deal with the exact distribution of T. In
developing the sampling distribution of T, we shall assume that our random
sample was selected from a normal population. We can then write
where
and
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling
distributions of S2
When σ is
Unknown: t
distribution
31
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Unknown: t distribution
32
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Unknown: t distribution
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of S2
When σ is Unknown: t distribution
Ex.:
A chemical engineer claims that the population mean yield of a certain batch process is
500 grams per milliliter of raw material. To check this claim he samples 25 batches
each month. If the computed t-value falls between −t0.05 and t0.05, he is satisfied with
this claim. What conclusion should he draw from a sample that has a mean 𝑥 = 518
grams per milliliter and a sample standard deviation s=40 grams? Assume the
distribution of yields to be approximately normal.
Solution:
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of two sample variances
(F distribution)
35
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of two sample variances
(F distribution)
36
Lecture #2------Tuğba Efendigil, Ph.D.
Sampling distributions of two sample variances
(F distribution)
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Lecture #2------Tuğba Efendigil, Ph.D.
Sampling
distributions of
two sample
variances
(F distribution)
38
Sampling distributions of two sample variances
(F distribution)
Variance ratio distribution
39
Lecture #2------Tuğba Efendigil, Ph.D.