Sampling Distributions : Sampling Distributions
Concepts Covered
Basic concept of sampling distribution
Usage of sampling distributions
Central limit theorem
Application of Central limit theorem
Introduction
As a task of statistical inference, we usually follow the following steps:
Introduction
As a task of statistical inference, we usually follow the following steps:
Data collection
Collect a sample from the population.
Introduction
As a task of statistical inference, we usually follow the following steps:
Data collection
Statistics
Collect a sample from the population.
Compute a statistics from the sample.
Introduction
As a task of statistical inference, we usually follow the following steps:
Data collection
Collect a sample from the population.
Statistics
Compute a statistics from the sample.
Statistical
inference
From the statistics we make various statements concerning the
values of population parameters.
For example, population mean from the sample mean, etc.
Basic Terminologies
Some basic terminology which are closely associated to the above-mentioned tasks
are reproduced below.
Basic Terminologies
Some basic terminology which are closely associated to the above-mentioned tasks
are reproduced below.
Population
A population consists of the totality of the observation, with
which we are concerned.
Basic Terminologies
Some basic terminology which are closely associated to the above-mentioned tasks
are reproduced below.
Population
A population consists of the totality of the observation, with which we are
concerned.
Sample
A sample is a subset of a population.
Basic Terminologies
Some basic terminology which are closely associated to the above-mentioned tasks
are reproduced below.
Population
Sample
Statistical
inference
A population consists of the totality of the observation, with which we are
concerned.
A sample is a subset of a population.
It is an analysis basically concerned with generalization and
prediction.
Statistical Inference
There are two facts, which are key to statistical inference
Statistical Inference
There are two facts, which are key to statistical inference
1.
Population parameters are fixed number whose values are usually unknown.
Statistical Inference
There are two facts, which are key to statistical inference
1.
Population parameters are fixed number whose values are usually unknown.
2.
Sample statistics are known values for any given sample, but vary from sample to
sample, even taken from the same population.
Statistical Inference
There are two facts, which are key to statistical inference
1.
2.
Population parameters are fixed number whose values are usually unknown.
Sample statistics are known values for any given sample, but vary from sample to sample, even taken
from the same population.
It is unlikely for any two samples drawn independently, producing identical values of
sample statistics.
Statistical Inference
There are two facts, which are key to statistical inference
1.
2.
Population parameters are fixed number whose values are usually unknown.
Sample statistics are known values for any given sample, but vary from sample to sample, even taken
from the same population.
It is unlikely for any two samples drawn independently, producing identical values of sample statistics.
The variability of sample statistics is always present and must be accounted for in any
inferential procedure
Statistical Inference
There are two facts, which are key to statistical inference
1.
2.
Population parameters are fixed number whose values are usually unknown.
Sample statistics are known values for any given sample, but vary from sample to sample, even taken
from the same population.
It is unlikely for any two samples drawn independently, producing identical values of sample statistics.
The variability of sample statistics is always present and must be accounted for in any inferential procedure
This variability is called sampling variation.
Statistical Inference
There are two facts, which are key to statistical inference
1.
2.
Population parameters are fixed number whose values are usually unknown.
Sample statistics are known values for any given sample, but vary from sample to sample, even taken
from the same population.
It is unlikely for any two samples drawn independently, producing identical values of sample statistics.
The variability of sample statistics is always present and must be accounted for in any inferential procedure
This variability is called sampling variation.
Note:
A sample statistics is random variable and like any other random variable, a sample
statistics has a probability distribution.
Sampling Distributions
A sample statistics is random variable and like any other random variable, a sample
statistics has a probability distribution. The distribution is used to describe the
variability of sample statistics.
Sampling Distributions
A sample statistics is random variable and like any other random variable, a sample statistics has a
probability distribution. The distribution is used to describe the variability of sample statistics.
Definition: Sampling Distribution
The sampling distribution of a statistics is the probability distribution of that statistics.
Sampling Distributions
A sample statistics is random variable and like any other random variable, a sample statistics has a
probability distribution. The distribution is used to describe the variability of sample statistics.
Definition: Sampling Distribution
The sampling distribution of a statistics is the probability distribution of that statistics.
തis
The probability distribution of sample mean (hereafter, will be denoted as 𝑋
called the sampling distribution of the mean (also, referred to as the distribution of
sample mean).
Sampling Distributions
A sample statistics is random variable and like any other random variable, a sample statistics has a
probability distribution. The distribution is used to describe the variability of sample statistics.
Definition: Sampling Distribution
The sampling distribution of a statistics is the probability distribution of that statistics.
തis called the sampling
The probability distribution of sample mean (hereafter, will be denoted as 𝑋
distribution of the mean (also, referred to as the distribution of sample mean).
ത, we call sampling distribution of variance (denoted as 𝑆2)
Like 𝑋
Sampling Distributions
A sample statistics is random variable and like any other random variable, a sample statistics has a
probability distribution. The distribution is used to describe the variability of sample statistics.
Definition: Sampling Distribution
The sampling distribution of a statistics is the probability distribution of that statistics.
തis called the sampling
The probability distribution of sample mean (hereafter, will be denoted as 𝑋
distribution of the mean (also, referred to as the distribution of sample mean).
ത, we call sampling distribution of variance (denoted as 𝑆2)
Like 𝑋
തand 𝑆2 for different random samples of a population, we are
Using the values of 𝑋
to make inference on the parameters 𝜇 and 𝜎2(of the population).
Issues with Sampling distribution
In practical situation, for a large population, it is infeasible to have all
possible samples and hence probability distribution of sample statistics.
Issues with Sampling distribution
In practical situation, for a large population, it is infeasible to have all
possible samples and hence probability distribution of sample statistics.
The sampling distribution of a statistics depends on
Issues with Sampling distribution
In practical situation, for a large population, it is infeasible to have all
possible samples and hence probability distribution of sample statistics.
The sampling distribution of a statistics depends on
the type of the population
Issues with Sampling distribution
In practical situation, for a large population, it is infeasible to have all
possible samples and hence probability distribution of sample statistics.
The sampling distribution of a statistics depends on
the type of the population
the size of the samples, and
Issues with Sampling distribution
In practical situation, for a large population, it is infeasible to have all
possible samples and hence probability distribution of sample statistics.
The sampling distribution of a statistics depends on
the type of the population
the size of the samples, and
the method of choosing the samples.
Sampling Distribution−Example 1
Problem
Consider five identical disks numbered as 1, 2, 3, 4 and 5. Consider an experiment consisting of drawing
two disks, replacing the first before drawing the second, and then computing the mean of the values of the
two disks.
Sampling Distribution−Example 1
Solution
Following table lists all possible samples and their means
Sample(X)
Mean(𝑿ഥ)
Sample(X)
Mean(𝑿ഥ)
Sample(X)
Mean(𝑿ഥ)
[1,1]
1.0
[2,5]
3.5
[4,4]
4.0
[1,2]
1.5
[3,1]
2.0
[4,5]
4.5
[1,3]
2.0
[3,2]
2.5
[5,1]
3.0
[1,4]
2.5
[3,3]
3.0
[5,2]
3.5
[1,5]
3.0
[3,4]
3.5
[5,3]
4.0
[2,1]
1.5
[3,5]
4.0
[5,4]
4.5
[2,2]
2.0
[4,1]
2.5
[5,5]
5.0
[2,3]
2.5
[4,2]
3.0
[2,4]
3.0
[4,3]
3.5
Consider five identical disks numbered as 1,
2, 3, 4 and 5. Consider an experiment
consisting of drawing two disks, replacing
the first before drawing the second, and
then computing the mean of the values of
the two disks.
Sampling Distribution−Example 1 contd…
Solution
Sampling Distribution of means
ത
𝐱
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
f(𝑥ҧ
)
1
25
2
25
3
25
4
25
5
25
4
25
3
25
2
25
1
25
1. 0
1.5
2. 0
2.5
3. 0
3. 5
4.0
4. 5
5. 0
Consider five identical disks numbered as 1,
2, 3, 4 and 5. Consider an experiment
consisting of drawing two disks, replacing
the first before drawing the second, and
then computing the mean of the values of
the two disks.
Sampling Distribution−Example 1 contd…
Solution
Sampling Distribution of means
The distribution closely
resembles a normal
distribution
than
a
uniform distribution.
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Consider five identical disks numbered as 1,
2, 3, 4 and 5. Consider an experiment
consisting of drawing two disks, replacing
the first before drawing the second, and
then computing the mean of the values of
the two disks.
Sampling Distribution−Example 1 contd…
Solution
Sampling Distribution of means
The distribution closely
resembles a normal
distribution
than
a
uniform distribution.
The mean of the
distribution of 𝑥ҧvalues is
3 and the variance is 1.
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Consider five identical disks numbered as 1,
2, 3, 4 and 5. Consider an experiment
consisting of drawing two disks, replacing
the first before drawing the second, and
then computing the mean of the values of
the two disks.
Theorem on Sampling Distribution
Theorem: Sampling Distribution of Mean
ത from a random sample of size 𝑛 drawn from a
The sampling distribution of 𝑋
2
ത= 𝜇 and variance 𝑆2 = 𝜎
population with mean 𝜇 and variance 𝜎 2 will have mean 𝑋
𝑛
Theorem on Sampling Distribution
Theorem: Sampling Distribution of Mean
ത from a random sample of size 𝑛 drawn from a population with mean 𝜇 and
The sampling distribution of 𝑋
2
ത= 𝜇 and variance 𝑆2 = 𝜎
variance 𝜎 2 will have mean 𝑋
𝑛
With reference to data in Example 1 (considering Uniform distribution)
Theorem on Sampling Distribution
Theorem: Sampling Distribution of Mean
ത from a random sample of size 𝑛 drawn from a population with mean 𝜇 and
The sampling distribution of 𝑋
2
ത= 𝜇 and variance 𝑆2 = 𝜎
variance 𝜎 2 will have mean 𝑋
𝑛
With reference to data in Example 1 (considering Uniform distribution)
For the population, 𝜇 =
5+1
=3
2
Theorem on Sampling Distribution
Theorem: Sampling Distribution of Mean
ത from a random sample of size 𝑛 drawn from a population with mean 𝜇 and
The sampling distribution of 𝑋
2
ത= 𝜇 and variance 𝑆2 = 𝜎
variance 𝜎 2 will have mean 𝑋
𝑛
With reference to data in Example 1 (considering Uniform distribution)
For the population, 𝜇 = 5+1 = 3
2
𝜎 2 = 25−1 = 2
12
Applying the theorem, we have 𝑋ത=3𝑎𝑛𝑑 𝑆2=1
Hence, the theorem is verified!
Central Limit Theorem
The Theorem on sampling distribution is also true if we sample from a population
തwill be approximately
with unknown distribution, the sampling distribution of 𝑋
normal with mean μ and variance
𝜎2
𝑛
Central Limit Theorem
The Theorem on sampling distribution is also true if we sample from a population with unknown
തwill be approximately normal with mean μ and variance 𝜎
distribution, the sampling distribution of 𝑋
2
𝑛
This further, can also be established with the famous “central limit theorem”,
which is stated below.
Central Limit Theorem
The Theorem on sampling distribution is also true if we sample from a population with unknown
തwill be approximately normal with mean μ and variance 𝜎
distribution, the sampling distribution of 𝑋
2
𝑛
This further, can also be established with the famous “central limit theorem”, which is stated below.
Theorem: Central Limit Theorem
If random samples each of size 𝑛 are taken from any distribution with mean 𝜇 and
തwill have a distribution approximately normal
variance 𝜎 2 , the sample mean 𝑋
𝜎2
with mean 𝜇 and variance
𝑛
Central Limit Theorem
The Theorem on sampling distribution is also true if we sample from a population with unknown
തwill be approximately normal with mean μ and variance 𝜎
distribution, the sampling distribution of 𝑋
2
𝑛
This further, can also be established with the famous “central limit theorem”, which is stated below.
Theorem: Central Limit Theorem
If random samples each of size 𝑛 are taken from any distribution with mean 𝜇 and variance 𝜎 2 , the
തwill have a distribution approximately normal with mean 𝜇 and variance 𝜎
sample mean 𝑋
2
𝑛
The approximation becomes better as 𝒏 increases.
Applicability of Central Limit Theorem
1
The theorem is an asymptotic result (being exactly true only if n goes to infinity),
however the approximation is usually very good for quite moderate values of n.
Sample sizes required for the approximation to be useful depend on the nature
of the distribution of the population.
2
5
6
3
For populations that resemble the normal, sample sizes of 10 or more are
usually sufficient
4
Sample sizes in excess of 30 are adequate for virtually all populations, unless
the distribution is extremely skewed
If the population is normally distributed, the sampling distribution of the mean
is exactly normally distributed regardless of sample size
Finally, one very important application of the Central Limit Theorem is the
determination of reasonable values of the population mean 𝜇
REFERENCES
Probability and Statistics for
Engineers and Scientists (8th Ed.)
by Ronald E. Walpole, Sharon L.
Myers, Keying Ye (Pearson), 2013.
Statistics, Witte, R.S., Witte, J.S.,
2013.. Wiley
Probability and Statistics, 2nd
edition, Moris DeGroot, 1986.
Reading, MA: Addison-Wesley.
Freund, R.J. & Wilson, W.J.
Statistical Methods, Academic
Press, San Diego, 1997
Sampling Distributions
Concepts Covered
Basic concept of sampling distribution
Usage of sampling distributions
Central limit theorem
Application of Central limit theorem
Applicability of Central Limit Theorem
1
The theorem is an asymptotic result (being exactly true only if n goes to infinity),
however the approximation is usually very good for quite moderate values of n.
Sample sizes required for the approximation to be useful depend on the nature
of the distribution of the population.
2
5
6
3
For populations that resemble the normal, sample sizes of 10 or more are
usually sufficient
4
Sample sizes in excess of 30 are adequate for virtually all populations, unless
the distribution is extremely skewed
If the population is normally distributed, the sampling distribution of the mean
is exactly normally distributed regardless of sample size
Finally, one very important application of the Central Limit Theorem is the
determination of reasonable values of the population mean 𝜇
Usefulness of the Sampling Distribution
The mean of the sampling distribution of the mean is the population mean.
This implies that “on the average” the sample mean is the same as the population mean.
Usefulness of the Sampling Distribution
The mean of the sampling distribution of the mean is the population mean.
This implies that “on the average” the sample mean is the same as the population mean.
We therefore say that the sample mean is an unbiased estimate of the population mean.
Usefulness of the Sampling Distribution
The mean of the sampling distribution of the mean is the population mean.
This implies that “on the average” the sample mean is the same as the population mean.
We therefore say that the sample mean is an unbiased estimate of the population mean.
The variance of the distribution of the sample means is
𝜎2
𝑛
The standard deviation of the sampling distribution (i.e., 𝜎ൗ
often called the standard error of the mean.
𝑛 ) of the mean,
Usefulness of the Sampling Distribution
The mean of the sampling distribution of the mean is the population mean.
This implies that “on the average” the sample mean is the same as the population mean.
We therefore say that the sample mean is an unbiased estimate of the population mean.
The variance of the distribution of the sample means is
𝜎2
𝑛
The standard deviation of the sampling distribution (i.e., 𝜎ൗ
often called the standard error of the mean.
𝑛 ) of the mean,
If σ is high then the sample mean are not reliable, however for a very large sample size
(𝑛→∞), standard error tends to zero
Example−2
Problem
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.
Example−2
Problem
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
Here, a sample of 16 bulbs is drawn from the population.
𝑆𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛 = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛 = 800 ℎ𝑟𝑠
Example−2
Problem
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
Here, a sample of 16 bulbs is drawn from the population.
𝑆𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛 = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛 = 800 ℎ𝑟𝑠
𝑆. 𝐷 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 = 𝑆.𝐷 𝑜𝑓 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒
= 𝜎
𝑛
= 40
16
= 10
Example−2
Problem
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
Here, a sample of 16 bulbs is drawn from the population.
𝑆𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛 = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛 = 800 ℎ𝑟𝑠
𝑆. 𝐷 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 = 𝑆.𝐷 𝑜𝑓 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒
= 𝜎
𝑛
= 40
16
= 10
P(average life of given sample< 𝟕𝟕𝟓 )
= 𝑃 𝑥ҧ< 775
= 𝑃 𝑧 < −2.5
= 𝟎. 𝟎𝟎𝟔𝟐
Sampling Distribution
Theorem: Sampling Distribution of the Difference Between Two Means
If independent samples of size 𝑛1 and 𝑛2 are drawn at random from two
populations, discrete or continuous, with means 𝜇1 and 𝜇2 and variances 𝜎 21 and 𝜎22,
respectively, then the sampling distribution of the differences of means, 𝑋1 − 𝑋2, is
approximately normally distributed with mean and variance given by
𝜎2
𝑛1
𝜎2
𝑛2
𝜇𝑋ത1 −𝑋ത2 = 𝜇1 − 𝜇2 and 𝜎2𝑋ത1 −𝑋ത2 = 1 + 2
Sampling Distribution
Theorem: Sampling Distribution of the Difference Between Two Means
If independent samples of size 𝑛1 and 𝑛2 are drawn at random from two populations, discrete or
continuous, with means 𝜇1 and 𝜇2 and variances 𝜎12 and 𝜎 22, respectively, then the sampling distribution of
the differences of means, 𝑋1 − 𝑋2, is approximately normally distributed with mean and variance given by
𝜎2
𝜎2
𝑛1
𝑛2
𝜇𝑋ത1−𝑋ത2 = 𝜇1 − 𝜇2 and 𝜎2 𝑋ത1−𝑋ത2 = 1 + 2
Hence,
𝑍=
᪄ 1−𝑋
᪄ 2 − 𝜇 1 −𝜇 2
𝑋
𝜎 2 /𝑛 1 + 𝜎 2 /𝑛 2
1
2
Sampling Distribution
Theorem: Sampling Distribution of the Difference Between Two Means
If independent samples of size 𝑛1 and 𝑛2 are drawn at random from two populations, discrete or
continuous, with means 𝜇1 and 𝜇2 and variances 𝜎12 and 𝜎 22, respectively, then the sampling distribution of
the differences of means, 𝑋1 − 𝑋2, is approximately normally distributed with mean and variance given by
𝜎2
𝜎2
𝑛1
𝑛2
𝜇𝑋ത1−𝑋ത2 = 𝜇1 − 𝜇2 and 𝜎2 𝑋ത1−𝑋ത2 = 1 + 2
Hence,
𝑍=
᪄ 1−𝑋
᪄ 2 − 𝜇 1 −𝜇 2
𝑋
𝜎 2 /𝑛 1 + 𝜎 2 /𝑛 2
1
2
𝒁 is approximately a standard normal variable.
Sampling Distribution contd…
Theorem: Sampling Distribution of the Difference Between Two Means (contd…)
Reproductive property of normal distribution: If 𝑋1, 𝑋2, … … , 𝑋𝑛 are independent
random variables, having normal distribution with mean μ1, μ2, … … , μ𝑛 and variance
𝜎12 , 𝜎22,……, 𝜎𝑛2 then the random variable 𝑋
= 𝑎 𝑋1 1 + 𝑎2 𝑋2 +……+𝑎 𝑛 𝑋𝑛 has normal
distribution with
mean, 𝜇𝑋ത = 𝑎1𝜇1 + 𝑎2𝜇2 + …… + 𝑎𝑛𝜇 𝑛
variance, 𝜎 2 = 𝑎 2𝜎 2 + 𝑎 2𝜎 2 + …… + 𝑎 2𝜎 2
ത
𝑋
1
1
2
2
𝑛
𝑛
Example−3
Problem
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(𝑋ത𝐴−𝑋ത𝐵 > 1.0), where
𝑋ത𝐴and 𝑋ത𝐵are average drying times for samples of size 18.
Example−3
Solution
From the sampling distribution of 𝑋𝐴 − 𝑋𝐵, we know that the
distribution is approximately normal with mean
𝜇 𝑋𝐴−𝑋𝐵 = 𝜇𝐴 − 𝜇𝐵 = 0 and variance
𝜎 𝐴2
𝜎 𝐵2
2
𝜎𝑋 −𝑋 = + = 1 + 1 = 1
𝐴
𝐵
𝑛𝐴
𝑛𝐵
18
18
9
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(𝑋ത𝐴 −𝑋ത𝐵 >
1.0), where 𝑋ത𝐴 and 𝑋ത𝐵 are average drying
times for samples of size 18.
Example−3
Solution
From the sampling distribution of 𝑋𝐴 − 𝑋𝐵, we know that the
distribution is approximately normal with mean
𝜇 𝑋 𝐴 −𝑋 𝐵 = 𝜇𝐴 − 𝜇𝐵 = 0 and variance
𝜎2
𝑋 𝐴 −𝑋 𝐵
2
2
𝑛𝐴
𝑛𝐵
= 𝜎 𝐴 + 𝜎𝐵 = 1 + 1 = 1
18
18
9
᪄𝐴 − 𝑋
᪄ 𝐵 = 1.0,
Corresponding to the value 𝑋
we have
𝑧 = 1− 𝜇 𝐴 −𝜇 𝐵 = 1−0 = 3.0
1/9
1/9
𝑃 𝑍 > 3.0 = 1 − 𝑃 𝑍 < 3.0
= 1 − 0.9987
= 𝟎. 𝟎𝟎𝟏𝟑
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(𝑋ത𝐴 −𝑋ത𝐵 >
1.0), where 𝑋ത
𝐴 and 𝑋ത
𝐵 are average drying
times for samples of size 18.
CONCLUSION
In this lecture we learned the basics of sampling distribution.
Next, we learned three very important theorems of statistics:
Theorem on sampling distribution
Central limit theorem
Sampling distribution of the difference between two means
Learners are instructed to understand these theorems theoretically
and also via practice problems.
In the next lecture, we will look into more concepts of Sampling
distribution.
REFERENCES
Probability and Statistics for
Engineers and Scientists (8th Ed.)
by Ronald E. Walpole, Sharon L.
Myers, Keying Ye (Pearson), 2013.
Statistics, Witte, R.S., Witte, J.S.,
2013.. Wiley
Probability and Statistics, 2nd
edition, Moris DeGroot, 1986.
Reading, MA: Addison-Wesley.
Freund, R.J. & Wilson, W.J.
Statistical Methods, Academic
Press, San Diego, 1997