This is about Sampling and Sampling Distribution

1. Business Statistic-II
Sampling and Sampling
Distribution
March 24, 2016

2. Introduction
In statistical inference use random sample or samples
to extract information about the population from
which it is drawn. The information extract is in the
form of summary statistic: a sample mean, a sample
standard deviation or other measures computed from
sample. Sample statistic are treated as estimator of
population parameters - µ, σ, ƥ etc.

3. Sampling
It is the process of selecting a sample from a
population is called sampling.
In sampling, a representative sample or portion
of elements of a population or process in selected
and analyzed . Based on sample results, called
sample statistic, statistical inferences, are made
about the population characteristic.

4. Reasons of Sample Survey
1. Movement of population element.
2. Cost and/or time required to contact the
whole population.
3. Destructive nature of certain test.

5. Sampling and Non-sampling
Errors
Sampling error is the deviation of the selected
sample from the true characteristics, traits,
behaviors, qualities or figures of the entire
population.
It occurs because researchers draw
different subjects from the same population but
still, the subjects have individual differences.

6. Non-sampling arises from inaccurate sampling
frame, data clarification or verification methods,
reporting or coding of data, and/or specifications. It
may also arise from poorly designed survey
questionnaires, improper sample allocation and
selection procedures, and/or errors in estimation
methodology.

7. Principle of Sampling
There are two important principle of sampling:
a) Principle of statistical regularity.
b) Principle of inertia of large
number.

8. Sampling Method
The represents basis and the element selection techniques
from the given population, classify several sampling method
into two categories:
Sampling Method
Probability
Non-
probability
1. Simple random sampling
2. Stratified sampling
3. Cluster sampling
4. Systematic sampling
5. Multi-stage sampling
1. Convenience sampling
2. Purposive sampling
3. Judgment sampling
4. Quota sampling

9. Sampling Distribution
In statistics, a sampling distribution or finite-sample
distribution is the probability distribution of a given
statistic based on a random sample. Sampling
distributions are important in statistics because they
provide a major simplification en route to statistical
inference

10. Sampling Distribution of Sample
Mean
In general, the sampling distribution of sample mean
depending on the distribution on the distribution of
the population or process from which samples are
drawn.

11. Sampling distribution of mean when
population has non-normal distribution
If population is not normally distribute, then we
make use if the central limit theorem to described
the random nature.
Central limit theorem a result that enables the use of
normal probability distribution to approximately the
sampling distribution of x̄ and p̄ .

12. Sampling distribution of mean when
population has normal distribution
Population standard deviation σ is known: No
matter what the population is for any given sample
of size n taken from a population with mean µ and
standard deviation σ, the sampling distribution of a
sample statistic, such as mean and standard
deviation are defined respectively by
• Mean of the distribution of sample means or
expected value of the mean µᵪ̵ or E(x̄ ) = µ
• Standard deviation (or error) of the distribution
of sample means or standard error of the mean.
𝜎ᵪ̵ =
𝜎
√𝑛

13. The value of sample mean x̄ is first covered into a
value z on the standard normal distribution to know
how any single mean value deviates from the mean
x̄ of sample mean values, by using the formula:
𝑧 =
𝑥−𝜇ᵪ̵
𝜎ᵪ̵
=
𝑥−µ
𝜎√𝑛

14. The procedure for making statistical inference using
sampling distribution about the population mean µ
based on mean x̄ on sample mean is summarised as
follows:
If the population standard deviation σ value in known and
either
Population distributing in normal
Population distribution is not normal but sample size n is
large (n≥ 30), then, the sampling distribution of mean µᵪ̵ = µ
and standard deviation σᵪ̵ = σ 𝑛, close to the standard
normal distribution given by
𝑧 =
𝑥−𝜇ᵪ̵
𝜎ᵪ̵
=
𝑥−µ
𝜎√𝑛

15. If the population size is finite with N elements
whose mean is µ and variance is σ² and samples of
fixed size n are drawn without replacement, then
the standard deviation (also called standard error)
of sampling distribution of mean x̄ can be modified
to adjust the continued change in the size of the
population N due to the several draws of samples of
the size n follows:
σᵪ̵ =
𝜎
√𝑛
𝑁−𝑛
𝑁−1

16. Population standard deviation σ is not known:
While calculating standard error σᵪ̵ of normally
distributed sampling distribution, so far we have
assumed that the population standard deviation σ is
known. However, if σ is not known, the value of the
normal viriate z cannot be calculated for a specific
sample.

17. In such a case, the standard deviation of population
σ must be estimated using sample standard
deviation σ. This the standard deviation error of the
sampling distribution of mean x̄ become
σᵪ̵ =
𝑥
√𝑛
Since the value of σᵪ̵ varies according to each
sample standard deviation therefore instead of
using the conversion formula:
z=
𝑥−µ
σ/√𝑛

18. Example: The mean length of life of a certain
cutting tools is 41.5 hours with a standard deviation
of 2.5 hours. What is the probability that a sample
random sample of size 50 drawn from this
population will have a mean between 40.5 hours
and 42 hours?

19. Solution: we are given the following information
µ=41.5 hours, σ=2.5 hours and n= 50
It is required to find the probability that the mean
length of life, x̄, of the cutting tools lies between
40.5 hours and 42 hours, that is P(40.5≤x≤̄42).
Base upon the given information, the statistic
sampling distribution are computed as:
µᵪ̵ = µ = 41.5
and σᵪ̵ =
𝜎
√𝑛
=
2.5
50
=
2.5
7.0711
= 03536

20. This population distribution is unknown, but sample
size n= 50 is large enough to apply the central limit
theorem. Hence, the normal distribution can be
used to find the required probability as known by
the shaded area in the following figure:

21. P(40.5≤x̄ ≤ 42) = P
𝑥1−𝜇
σᵪ
≤ 𝑧 ≤
x2−µ
σᵪ
=P
40.5−41.5
0.3536
≤ 𝑧 ≤
42−41.5
0.3536
=P −2.8281 ≤ 𝑧 ≤ 1.4140
=P 𝑧 ≥ −2.8281 + 𝑧 ≤ 1.4140
=P 0.4977+ 0.4207
= 0.9184
Thus 0.9184 is the probability of the tool of having
a mean life between the required hours.

22. Any Questions……………?
Please ask……………….
Thank you