Sampling and

Sampling
Distribution
 1 Illustrate random sampling

Distinguish between parameter and

OBJECTIVES 2 statistic

Identify sampling distribution of statistics

3 (sample mean)
Lesson 1.
Random
Sampling
 Population- it is a complete collection
of all elements to be studied.
DEFINITIO
N OF Census- It is a collection of data from
TERMS every element in a population

Sample- It is a sub-collection of
elements drawn from a population.
We want to know about these We have these to work with

Random
selection

Sample

Population

Inference
Parameter Statistic
Population Mean (Sample Mean)
5
Decide whether the statement describes a
parameter or statistic.
1. The average income of 40 out of 100 households in a
certain Barangay is P 12, 213.00 a month.
2. Percentage of red cars in the Philippines.
3. Number of senior high schools in Region 3.
4. A recent survey of a sample of 250 high school students
reported the average weight of 54.3 kg.
5. Average age of students in East High School.
Random Sampling
- an unbiased way in choosing respondents to
achieve valid and unbiased results of research

1. Simple Random Sampling

2. Systematic Random Sampling
3. Stratified random Sampling
4. Cluster or Area Random Sampling
Simple Random Sampling
- is a sampling A researcher wants to study the effects of
technique in which social media on Grade 11 students in
every element of the SFHS. He wishes to muse simple random
sampling technique in choosing the
population has the
members of his sample. If there are 1,000
same probability of Grade 11 students in the school, how many
being selected for students should there be in his sample?
inclusion in the Discuss the steps he must take if he wishes
sample to use the lottery method.
A researcher wants to study the effects of social media on Grade 11 students in TCSHS. He wishes to muse simple
random sampling technique in choosing the members of his sample. If there are 1,000 Grade 11 students in the
school, how many students should there be in his sample? Discuss the steps he must take if he wishes to use the
lottery method.
Systematic Random Sampling
- is a sampling technique in which In a group of 250
a list of elements to be included of students, how will
the population is used as a you select a sample
sampling frame and the elements containing 71
to be included in the desired students by using
sample are selected by skipping the systematic
through the list at a regular sampling
intervals technique?
In a group of 250 students, how will you select a sample containing 71 students by using the
systematic sampling technique?
Stratified Random Sampling
- is a sampling You want to interview 200 students in
your school to determine their opinion
technique in which the
on the new school uniform. How are
population is first you going to choose your sample by
divided into strata and using stratified sampling of there are
then samples are 1,200 students in grade 7; 1,100 in
grade 8; 1,050 in grade 9; 940 in grade
randomly selected
10; 900 in grade 11, and 810 in grade
separately from each 12?
stratum
You want to interview 200 students in your school to determine their opinion on the new school
uniform. How are you going to choose your sample by using stratified sampling of there are
1,200 students in grade 7; 1,100 in grade 8; 1,050 in grade 9; 940 in grade 10; 900 in grade 11,
and 810 in grade 12?
Cluster/Area Random Sampling
- is a sampling technique A researcher want to
in which the entire determine who among the
population is broken into families in small town are
small groups, or clusters, using the new detergent
and then some of the product. How is she going to
clusters are randomly do this using the cluster
selected sampling technique?
A researcher want to determine who among the families in small town are using the new
detergent product. How is she going to do this using the cluster sampling technique?
Assessment
Directions: Identify the type of sampling method.
1. The teacher writes all the names of students in a piece of
paper and puts it in a box for the graded recitation.
2. The teacher gets the class record and call every 4th name in the
list.
3. Every five files out of 500 files will be chosen.
4.There are 20 toddlers, 40 teenagers, 45 middle aged and 55
senior citizens in a certain area. Samples are taken according
to the total number of people in the area.
5. All the names of the employees of the company are put in
a rafl e box.
 Lesson 2.
Sampling Distribution of
the Sample Means
A sampling distribution of sample mean is a frequency
distribution using the means computed from all possible
random samples of a specific size taken from a population.

What’s In.
A population consists of the five numbers 2, 3, 6, 10, and 12.
Consider samples of size 2 that can be drawn from this population.
Sample Mean

2,3 2.5
Example 1: A population consists of the five numbers 2, 3, 6,
10 and 12. Consider samples of size 2 that can be drawn from
this population.
A. How many possible samples can be drawn?

To answer this, use the formula NCn (the number of N

objects taken n at a time), where N is the total population
and n is the sample to be taken out of the population,
In this case N= 5 and n= 2

5 C2 = 10 So, there are 10 possible samples to be drawn.

B. Construct the sampling distribution of sample means.
List all the possible outcome and get the mean of every sample.
sample Sample mean
2,3 2.5
10,12 11
2,6 4
Observe that the means vary
2,10 6
from sample to sample. Thus,
2,12 7 any mean based on the sample
3,10 6.5 drawn from a population is
3,6 4.5 expected to assume different
3,12 7.5 values for samples.
6,10 8
6,12 9
C. This time, let us make a probability distribution of the
sample means. This probability distribution is called the
sampling distribution of the sample means.
Sample mean Probability
2.5 1/10 or 0.1
11 1/10 or 0.1
4 1/10 or 0.1 Observe that all sample means
6 1/10 or 0.1
appeared only one; thus, their
7 1/10 or 0.1
probability is P(x)= 1 10 or 0.1
6.5 1/10 or 0.1
4.5 1/10 or 0.1
7.5 1/10 or 0.1
8 1/10 or 0.1
9 1/10 or 0.1
 Problem: Construct a sampling distribution of sample mean for the
set of data below.
86 88 90 95
98
Consider a sample size of 3 that can be drawn from a population.

A. How many possible samples can be drawn?

B. Construct the sampling distribution of sample means.

C.This time, let us make a probability distribution of the

sample means. This probability distribution is called, the
sampling distribution of the sample means.
A. How many possible samples can be drawn? In this
case N= 5 and n= 3 sample Sample mean
86, 88, 90 88
5C3 = 10 86, 90, 95 90

So, there are 10 possible samples to be drawn. 86, 90, 98 91

86, 90, 95 90
B. Construct the sampling
86, 90, 98 91
distribution of sample 86, 95, 98 93
means. 88, 90, 95 91

List all the possible 88, 90, 98 92

outcome and get 88, 95, 98 94

the mean of every 90, 95, 98 94

sample.
C. This time, let us make a probability distribution of the sample
means. This probability distribution is called, the sampling
distribution of the sample means.
Sample Mean Probability

88 1/10 or 0.1
Observe that 88, 92 and 93 appeared only
90 2/10 or 0.2 once; thus their probability is P(x)= 1/10 or
0.1. Since 90 and 94 appeared twice, their
91 3/10 or 0.3 probability is P(x)= 2/10 or 0.2. While 91
92 1/10 or 0.1 appeared thrice, their probability is P(x)=
3/ 10 or 0.3
93 1/10 or 0.1

94 2/10 or 0.2 Observe that the total probability of all sample means
must be equal to 1.
Assessment:
Direction : Find the Sample mean and probability of the following data

Construct all random samples consisting three

observations from the given data.
Arrange the observations in ascending order without
replacement and repetition.

86 89 92 95
98
 Lesson 3.
Finding the Mean and
Variance of the Sampling
Distribution of Means
 Objectives
At the end of the lesson, you are expected to:

1. Find the mean and variance of the sampling

distribution of the sample means
2. Define the sampling distribution of the sample mean
for normal population when the variance is: (a)
known (b) unknown
POPULATION
POPULATION MEAN

POPULATION VARIANCE
 Example
Consider a population consisting of 1, 2, 3, 4,
and 5. What is the mean and variance of the
population?
 Solution
POPULATION MEAN

The mean of the population is 3.00.

 Solution
POPULATION VARIANCE
X
1 -2 4 𝝈
𝟐
=
∑ ( 𝑿 −𝝁 ) 𝟐
=
𝟏𝟎
=𝟐
2 -1 1 𝑵 𝟓
3 0 0
4 1 1 The population variance is
5 2 4 equal to 2.
 Example
Consider a population consisting of 1, 2, 3, 4,
and 5. Suppose samples of size 2 are drawn
from this population. What is the mean and
variance of the sampling distribution of the
sample means?
 Solution
Determine the number of possible
samples of size 2.
NCn =
5C2

There are 10 possible samples of size 2 that can be drawn.

 Solution Samples
1, 2
Mean
1.50
List all possible 1, 3 2.00
samples and their 1, 4 2.50
1, 5 3.00
corresponding
2, 3 2.50
probabilities. 2, 4 3.00
2, 5 3.50
3, 4 3.50
3, 5 4.00
4, 5 4.50
 Solution
Construct the Sample Mean Frequency Probability
X P(x)
sampling 1.50 1 1/10
distribution of the 2.00 1 1/10
sample means. 2.50 2 2/10
3.00 2 2/10
3.50 2 2/10
4.00 1 1/10
4.50 1 1/10
Total 10 1.00
 Compute for the mean and
Solution variance of the sampling
distribution.
X P(X)

1.50 1/10 0.15 -1.50 2.25 0.225

2.00 1/10 0.20 -1.00 1 0.1
2.50 2/10 0.50 -0.50 0.25 0.05
3.00 2/10 0.60 0.00 0 0
3.50 2/10 0.70 0.50 0.25 0.05
4.00 1/10 0.40 1.00 1 0.1
4.50 1/10 0.45 1.50 2.25 0.225
 Properties of Sampling Distribution
of Sample Mean
If all possible samples of size n are drawn from a population of
size N with mean and variance , then the sampling
distribution of the sample means has the following properties:
1. The mean of the sampling distribution of the sample
means is always equal to the mean of the population.
 Properties of Sampling Distribution
of Sample Mean
2. The variance of the sampling
distribution of the sample means is
given by:
for finite population; and
for infinite population
 Properties of Sampling
Distribution of Sample Mean
FINITE POPULATION
- consists of a finite of fixed number of elements, measurements or
observations.

INFINITE POPULATION
- contains hypothetically at least, infinitely elements.
 Properties of Sampling
Distribution of Sample Mean
3. The standard deviation of the sampling
distribution of the sample means is given by:
for finite population
for infinite population
 Properties of Sampling
Distribution of Sample Mean
FINITE POPULATION
CORRECTION FACTOR

The finite population correction factor (fpc) is a mathematical

adjustment used when you're sampling from a small, known group of
items (finite population).
 Example
Consider a population consisting of 1, 2, 3, 4, and 5.
Suppose samples of size 2 are drawn from this
population. What is the mean and variance of the
sampling distribution of the sample means?
 Solution
POPULATION MEAN

The mean of the population is 3.00.

 Solution
POPULATION VARIANCE
X
1 -2 4 𝝈
𝟐
=
∑ ( 𝑿 −𝝁 ) 𝟐
=
𝟏𝟎
=𝟐
2 -1 1 𝑵 𝟓
3 0 0
4 1 1 The population variance is
5 2 4 equal to 2.
Solution

The variance of the

sampling distribution is
equal to 0.75.
 Standard Deviation of the
Sample Means
The standard deviation of the sampling distribution
of the sample means is also known as the standard
error of the mean. It measures the degree of accuracy
of the sample mean as an estimate of the population
mean .
 Good vs Poor Estimates
A good estimate of the mean is obtained if
the standard error of the mean is small
or close to zero, while a poor estimate, if
the standard error of the mean is large.
 Central Limit Theorem
If we want to get a good estimate of the
population mean, we have to make n
sufficiently large. This fact is stated as a
theorem, which is known as the Central
Limit Theorem.
 INFINITE POPULATION
Example 1:
A population has a mean of 60 and a standard
deviation of 5. A random sample of 16
measurements is drawn from this population.
Describe the sampling distribution of the sample
means by computing its mean and standard
deviation.
 Solution
Given:
, , and
Compute for the mean:

Compute for the standard deviation:

ACTIVITY
A. Consider the population below.
5 6 8 10 12 13
◦ 1. Compute the mean and the standard deviation of the population.
◦ 2. How many samples of size 5 can be generated from the given population?
◦ 3. Compute for the means of the sampling distribution of the sample means.
◦ 4. Calculate the variance and standard deviation of the sampling distribution
of the sample means.
B. The scores of individual students on a national test have a normal distribution
with mean 18.6 and standard deviation 5.9. At San Francisco High School, 76
students took the test. If the scores at this school have the same distribution as
national scores, what are the mean and standard deviation of the sample mean for
76 students?
LESSON 4
SOLVING PROBLEMS
INVOLVING SAMPLING
DISTRIBUTION OF THE
SAMPLE MEANS
 Objectives
At the end of the lesson, you are expected to:
1. Illustrate the Central Limit Theorem
2. Solve problems involving sampling distribution of
the sample means
 Central Limit Theorem
This theorem applies automatically to sampling from
infinite population. It also assures us that no matter what
the shape of the population distribution of the mean is, the
sampling distribution of the sample means is closely
normally distributed whenever n is large.
As n increases, the shape of the sampling
distribution of the sample means becomes wider
and more normal.
Central Limit Theorem

where = sample mean

= population mean
= population standard deviation
n = sample size
 Z-scores
z-score for individual scores

z-score for sample means

 Example
The average time it takes a group of
college students to complete a certain
examination is 46.2 minutes. The
standard deviation is 8 minutes. If 50
randomly selected college students take
the examination, what is the probability
that the mean time it takes the group to
complete the test will be less than 43
minutes?
 Example
Does it seem reasonable that a college
student would finish the examination in
less than 43 minutes?

Does it seem reasonable that the mean of

the 50 college students could be less than
43 minutes?
 Example
The average number of milligrams (mg) of
cholesterol in a cup of certain brand of ice
cream is 660 mg, and the standard
deviation is 35 mg. Assume the variable is
normally distributed.
a. If a cup of ice cream is selected, what is
the probability that the cholesterol
content will be more than 670 mg?
 Example
b. If a sample of 10 cups of ice cream is
selected, what is the probability that the
mean of the sample will be larger than
670 mg?
 Activity
A manufacturer of light bulbs
produces light bulbs that last a mean
of 950 hours with standard deviation
of 120 hours. What is the probability
that the mean lifetime of a random
sample of 10 of these bulbs is less
than 900 hours?