STATISTICS FOR MANAGEMENT-II
LEARNING OBJECTIVES:
 To take sample from an entire population and use it to describe the
population.
 Introduction to sampling and types of sampling.
 Introduction to sampling distribution and its concepts.
 To understand the trade-offs between the cost of taking larger
samples and the additional accuracy this gives to decision made from
them.
INSTRUCTOR’S NAME:
1
CHAPTER – 6
Parameters:
The observations taken from population used to calculate mean, median and
standard deviation etc. are called parameters.
For example, if we calculate mean from population, it is called population
mean and is denoted by µ.
Statistics:
The observations taken from sample used to calculate mean, median and
standard deviation etc .are called statistics.
For example, if we calculate mean from sample, it is called sample mean and
is denoted by x .
There are two types of population,
1-Finite Population:
Population having limited size is called finite population.
2- Infinite Population:
Population, in which it is theoretically impossible to observe all items.
There are two types of sampling,
1-Non-random or judgment sampling:
It is process in which personal judgment determine which units of population
are selected for sample.
2- Random or Probability Sampling:
It is a process in which each unit of population has a known (non zero)
probability of its being included in sample.
There are four methods for random sampling.
iiiiiiiv-
Simple Random Sampling.
Systematic Sampling.
Stratified Sampling.
Cluster Sampling.
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i-Simple Random Sampling:
It is method of selecting samples in which each possible sample has an equal
probability of being picked and each unit of population has an equal chance
of being included in the sample.
ii- Systematic Sampling:
It is method of selecting samples from population at uniform intervals that is
measured in time, order or space.
Iii-Stratified Sampling:
In this method we divide the population in to groups, called STRATA. Then
(a) We select sample from each stratum according to proportion of that
stratum in population.
(b) We draw an equal number of samples from each stratum and give weight
to result according to stratum’s proportion of total population.
We use stratified sampling when there is small variation in each group but
wide variation between groups (i.e. group to group)
iv- Cluster Sampling:
In this method we divide the population into groups, called CLUSTERS, and
then select a random sample of these cluster
We use cluster sampling when there is wide variation within each group but
small variation between groups (i.e. group to group).
EXERCISE:
(EX. Sc 6-1 pg 303)
List the number of elements selected for the data mention in the question,
based on the random digits table.
Solution:
0892
1652
2963 2913 3181 9348
4959
7695
7712 8136
9659
2526
6988 1781
2204
4339
6299 3397
7652 8559
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EXERCISE:
(EX. 6-11 pg 303)
A population is made up of groups that have wide variation within each
group but little variation from group to group. The appropriate type of
sampling for this population is,
(a) Stratified.
(b) Systematic.
(c) Cluster.
(d) Judgment.
Solution:
The cluster sampling is appropriate type of sampling for this population.
H.W:
Do
EX.6-10
pg 303.
Sampling with replacement:
A method in which sampled items are returned to the population after being
picked.
In this method, each item of population can appear in the sample more than
once.
Sampling Without Replacement:
A method in which sampled items are not returned to the population after
being picked.
In this method no item of population can appear in sample more than once.
Sampling Distribution:
A probability distribution of all values of a statistic (mean, median and
standard deviation etc.) is called sampling distribution.
There are many kinds of sampling distributions. Some are as normal
distribution, t-distribution and F-distribution etc.
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Sampling Distribution of mean:
A probability distribution of all the means of the samples (taken from
population) is called sampling distribution of mean. This distribution has its
own mean µ and standard deviation σ .
Standard Error:
Standard deviation of sampling distribution of statistic (mean, median and
standard deviation etc.) is known as standard error of that statistic (mean,
median and standard deviation etc.).
A sampling distribution of mean that has small standard error is better
estimator of population than sampling distribution mean that has larger
standard error.
Relationship between Sample Size and Standard Error:
As sample size increases the standard error decreases.
As the standard error decreases the value of any sample mean will probably
be closer to the value of population.
Sampling from normal population:
When population is normally distributed then,
=
σx̄
=
µ
𝜎
Here σx̄ is standard error of
√𝑛
mean from infinite population
and we sample with replacement.
σx̄
=
𝜎
√𝑛
𝑁−𝑛
× √𝑁−1
Here σx̄ is standard error of
mean from finite population
and we sample without replacement
Here
√
𝑁−𝑛
𝑁−1
is finite population multiplier. If sampling fraction
𝑛
𝑁
is less
than 0.05 then we will not use this multiplier.
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Sampling from non normal population:
Relationship between shape of sampling distribution of mean and shape of
population distribution is called the central limit theorem. It states as,
The sampling distribution of mean approaches to normal distribution, as the
sample size increases, regardless of the shape of population distribution .
Standard Normal Random Variable:
Z=
x −𝜇
σx̄
Here Z is standard normal random variable. With this we can determine the
probability that the sample mean will lie between the given limits.
EXERCISE:
(EX. Sc 6-3 pg 311)
Is the conclusion by quality control manager, with the information given in
question, correct?
Solution:
The conclusion is not correct as the mean of sample does not equal the
population mean because of sampling error.
EXERCISE:
(EX. 6-20 pg 311)
The term error, in standard error of the mean, refers to what type of error?
Solution:
The term error, in standard error of the mean refers to sampling error.
EXERCISE:
(EX. 6-22 pg 311)
Solution:
In general, over estimating the mean is neither better nor worse
than under estimating. In this case underestimate ($ 3.0) is closer to true
mean ($ 3.14) than overestimate ($ 3.5).So first sample is better one.
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EXERCISE:
(EX. Sc 6-5 pg 321)
Solution:
𝜇 = 98.6
𝜎 = 17.2
n = 25
(a)
σx̄ =
=
Z =
For
√𝑛
17.2
√25
=
17.2
5
= 3.44
x −𝜇
σx̄
x = 92
Z =
For
𝜎
x
92−98.6
3.44
=
−6.6
3.44
= -1.92
= 102
Z =
P(92 ≤
x
102−98.6
3.44
=
3.4
3.44
=
0.99
≤ 102) = P ( -1.92 ≤ Z ≤ 0.99)
= P (-1.92 ≤ Z ≤ 0) + P (0 ≤ Z ≤ 0.99)
= 0.4726 + O.3389
= 0.8115
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EXERCISE:
(EX. Sc 6-6 pg 321)
Solution:
𝜇 = 112
𝜎 = 56
n = 50
(a)
σx̄ =
=
Z =
For
𝜎
√𝑛
56
√50
56
=
7.07
= 7.92
x −𝜇
σx̄
x = 100
Z =
P ( x < 100)
100−112
7.92
=
−12
7.92
= -1.52
= P ( Z ≤ -1.52)
= P (-∞ ≤ Z ≤ 0) - P (-1.52 ≤ Z ≤ 0)
= 0.5 – O.4357
= 0.0643
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EXERCISE:
(EX. 6-28 pg 321)
Solution:
𝜇 = 18
𝜎 = 4.8
n = 19
(a)
σx̄ =
=
Z =
For
x = 16
Z =
For
x
𝜎
√𝑛
4.8
=
√19
4.8
= 1.10
4.36
x −𝜇
σx̄
16−18
1.10
=
−2
= -1.82
1.10
= 20
Z =
P(16 ≤
x
20−18
1.10
=
2
1.10
=
1.82
≤ 20) = P ( -1.82 ≤ Z ≤ 1.82)
= P (-1.82 ≤ Z ≤ 0) +P (0 ≤ Z ≤ 1.82)
= 0.4656 + O.4656
= 0.9312
H.W: Do EX. Sc 6-5 (b) Pg 321
EX. Sc 6-6 (b) Pg 321
EX. 6-28 (c) Pg 321
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EXERCISE:
(EX. Sc 6-7 pg 327)
Solution:
(𝑎)
𝜇 = 105
𝜎 = 17
n = 64
N = 125
σx̄ =
=
𝜎
√𝑛
17
√64
=
17
8
× √
𝑁−𝑛
𝑁−1
125−64
× √ 125−1
61
× √124
= 2.125 X √0.492
= 2.125 X 0.70
= 1.49
(b)
𝜇 = 105
𝜎 = 17
n = 64
N = 125
σx̄ = 1.49 ( already calculated)
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Z =
For
For
x −𝜇
σx̄
x = 107.5
x
Z =
107.5−105
1.49
=
Z =
109−105
1.49
=
2.5
1.49
= 1.68
= 109
P(107.5 ≤
x
4
1.49
=
2.68
≤ 109) = P ( 1.68 ≤ Z ≤ 2.68)
= P (0 ≤ Z ≤ 2.68) --P (0 ≤ Z ≤ 1.68)
= 0.4963 - O.4535
= 0.0428
EXERCISE:
(EX. 6-40 pg 327)
Solution:
(𝑎)
𝜇 = 364
𝜎 2 = 18
𝜎 = √18
n = 32
N = 75
σx̄ =
𝜎
√𝑛
= 4.24
𝑁−𝑛
× √ 𝑁−1
11
4.24
=
75−32
× √ 75−1
√32
4.24
43
5.66
× √74
= 0.75 X
√0.58
=
= 0.75 X 0.76
= 0.57
𝜎
σx̄ =
( c)
√𝑛
4.24
=
√32
=
4.24
5.66
= 0.75
H.W:
Do
EX. 6-40
Pg 327
EXERCISE:
(EX. 6-30 pg 321)
Solution:
e = |x
−
𝜇 = 375
𝜎 = 48
𝜇 |=5
P (Z) = 95%
= 0.95
=
0.95
2
= 0.4750
= 1.96
n =
[
𝑍𝑋 𝜎
]2
|x −𝜇|
12
1.96𝑋48 2
]
5
[
=
= (18.82)2
= =
[
94.08 2
]
5
= 354
EXERCISE:
(EX. Sc 6-8 pg 327)
Solution:
(𝑏)
e =|x
−
𝜇 = unknown
𝜎 = 1.25
𝜇 | = 0.5
P (Z) = 98%
= 0.98
=
0.98
2
= 0.4900
= 2.33
𝑍𝑋 𝜎
n =
[
=
[
2.33𝑋1.25 2
]
0.5
=
[
2.91 2
]
0.5
]2
|x −𝜇|
= (5.83)2
= 34
H.W:
If population has standard deviation $500.How many observation
would be needed in order to be 95% certain that sample mean is within
$100 of population mean?
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OBJECTIVE SECTION
Q-1 WRITE SHORT ANSWERS FOR THE FOLLOWING.
1-Write the name of the theorem which describes the relationship
between the shape of sampling distribution of mean and shape of
population distribution.
Answer:
Central Limit Theorem.
2- Define standard error.
Answer:
Standard deviation of sampling distribution of statistic (mean,
median etc) is known as standard error of that statistic (mean, median etc)
3- Define parameters.
Answer:
The observations taken from population are used to calculate
mean, median and standard deviation etc are called parameters.
4- Write the other name of random sampling.
Answer:
Probability sampling.
Q-2
TICK THE CORRECT ONE.
1- Standard deviation of sampling distribution of statistic is called
a)
Cluster
b)
Standard error
c)
Population
2- As the sample size increases the standard error ,
a)
Increases
b)
Decreases
c)
Remains same
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3- The other name of non-random sampling is,
a)
Systematic sampling
b)
Judgment Sampling
c)
Non of these
4- Standard error of sampling distribution of mean is denoted by,
Q-3
1-
a)
b)
N
c)
σx̄
FILL IN THE BLANKS:
In cluster sampling we divide the population in to groups called,
.
2- A sampling distribution of that has
estimator.
3-
standard error is the best
The observations taken from sample are called
.
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