CABT SHS Statistics & Probability - Sampling Distribution of Means

CABT Statistics & Probability – Grade 11 Lecture Presentation

Sampling Distribution
of Means
A Grade 11 Statistics and Probability Lecture

Population and Sample
Sampling Distribution of Means
A population is the set of all items or
individuals of interest.
Examples: All registered voters
Cell phones manufactured today
Sales receipts for August
A sample is a subset of the population.
Examples: 1200 voters selected at random for a
survey
A few cell phones selected for testing
Random receipts selected for audit

Inferential Statistics
Inferential statistics is concerned with
drawing conclusions and/or making decisions
concerning a population based only on
sample data.
http://www.gohomeworkhelp.com/admin/photos/what-is-inferential-statistics.jpg

Population inferences can be made...

...by selecting a representative sample from the
population

Why Sample?!
 Researchers use sampling if taking a
census of the entire population is
impractical.
 It is less time-consuming and less
costly than a census.
 It is possible that data from the sample
can be used to estimate corresponding
population measures.

Types of Sampling
Simple random sampling – every
element of the population has the same
probability of being selected for
inclusion in the sample.
Some simple random sampling
techniques:
1. Lottery or drawing – with or without
replacement
2. Using Random Numbers

Types of Sampling
Simple random sampling and Slovin’s
FormulaSlovin’s Formula is a formula used
to determine the sample size n from
a population of size N and margin of
error e is given by:

 2
1
N
n
Ne
Example: The number of samples needed for a population of
1,000 at a margin of error of 5% is
  
2
1000
286
1 1000 0.05
n  


Types of Sampling
– a list of randomly-arranged
elements of the population is
used as a sampling frame,
and the elements to be
included in the sample are
selected by skipping
through the list at regular
intervals.
Systematic random sampling
http://worldartsme.com/images/holiday-shopping-list-clipart-1.jpg

Types of Sampling
In doing systematic
random sampling in a
population of size N
using a sample size n,
the sampling interval k
is given by
Systematic random sampling
N
k
n

Example:
To get a sample
containing 8 students
from a group of 40
students using
systematic random
sampling, the sampling
interval is
40
5
8
k  

http://users.hist.umn.edu/~ruggles/hist5011/sampling.ppt

Types of Sampling
– the population is
partitioned into
several subgroups
or strata, then
samples are
randomly selected
separately from
each stratum
Stratified random sampling
https://faculty.elgin.edu/dkernler/statistics/ch01/images/strata-sample.gif

Types of Sampling
Proportional allocation of sample size
Stratified random sampling
Population
Strata
Number of
Students
per Strata
No. of
Samples
per Strata
Male 340 20
Female 500 30
TOTAL 840 50
340
50 20.24 20
840
males   
500
50 29.76 30
840
females   
Allocation of samples per
stratum:
Suppose that a sample of size 50 is to be obtained
from a population of 840 students grouped according
to gender:

Types of Sampling
– the population is
broken into small
groups or clusters,
then some of the
clusters are
randomly selected.
Cluster or area sampling
http://research-methodology.net/wp-content/uploads/2013/07/Cluster-sampling.gif

Types of Sampling
This type of
sampling is
frequently used
by Philippine
survey
companies like
SWS and Pulse
Asia.
Cluster or area sampling

Types of Sampling
The difference between cluster sampling and
stratified sampling is that the sample consists of
elements from the selected clusters only, while in
stratified sampling, the sample comes from all the
strata.
Cluster vs stratified sampling
http://keydifferences.com/wp-content/uploads/2016/04/stratified-vs-cluster-sampling.jpg

Your class would like to
interview 50 Junior High
School students to
determine their opinion
regarding the new school
ID. Explain how you can
choose your samples using
each of the sampling
techniques discussed.
Check your
understanding
https://boonesunriserotary.files.wordpress.com/2016/05/interview.jpg

Parameter & Statistic
A parameter is a descriptive
measure that describes a
population.A statistic is a descriptive measure
that describes a sample.
Usually, parameters are denoted by lower-
case GREEK letters (e.g.  or ), while
statistics use lower-case ROMAN letter (e.g. x
and s).

https://www.cliffsnotes.com/assets/267169.png

Parameter or statistic?
1. All CABT Senior High School students
2. A selected group of CABT Senior High
School students
3. Average scores of Grade 10 NAT takers
nationwide
4. Average of NAT scores taken from a
group of Grade 10 students from CABT

Determine if the given situation describes a
parameter or a statistic
Check your
understanding
1. A group of 20 high school students are
selected to join the Valenciana Festival.
2. Ten of the teachers in a SHS have Master’s
degrees.
3. The Philippine Senate has 24 senators.
4. CABT currently has 18 teachers.
5. Eight teachers in CABT are chosen to attend
a seminar.

Sampling Distributions
A sampling distribution is a probability
distribution of all of the possible values of a
statistic for a given sample size selected from
a population.
http://labs.geog.uvic.ca/geog226/images/Lab3/img_samp_dist.gif

Sampling Distribution of the Mean
A sampling distribution of
sample means is a frequency
distribution using the means
computed from all possible
random samples of a specific
size taken from a population.
The probability distribution of the sample
means is also called the sampling distribution
of the sample means, with the sample mean as
the random variable.
http://www.philender.com/courses/intro/notes2/sam.gif

Constructing Sampling Distributions of
Sample Means
How to Construct a Sampling Distribution of
Sample Means from a Given Population
STEP 1 – Determine the number of
samples of size n from the population of
size N.
STEP 2 – List all the possible samples
and compute the mean of each sample.
STEP 3 – Construct a frequency
distribution of the sample means
obtained in Step 2.
STEP 4 – Construct the probability

Sample Means
How to Construct a Sampling Distribution of
Sample Means from a Given Population
The number of samples of size n from the
population of size N are given as follows:
For sampling with
replacement:
For sampling without
replacement:
n
N
N n
C

Sample Means
Consider a group of N = 4 people
with the following ages: 18, 20, 22,
24.
Consider samples of size
n = 2 from the group. If X is
the age of one person in
the group and X is the
average age of the two
people in a sample,
construct the sampling
distribution of X.

Sample Means
Solution – sampling with replacement
Number of samples of size 2:
1st
2nd
Observation
Obs 18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
The 16 samples of size 2 The 16 sample means
2
4 16

Sample Means
Sample
Mean
Frequency Probability
18 1 1/16
19 2 1/8
20 3 3/16
21 4 1/4
22 3 3/16
23 2 1/8
24 1 1/16
TOTAL 16 1
X P(X)
18 1/16
19 1/8
20 3/16
21 1/4
22 3/16
23 1/8
24 1/16
Solution – sampling with replacement
2
4 16

Sample Means
Solution – sampling without
replacement
4 2
6C 
Sample
Sample
Mean
18, 20 19
18, 22 20
18, 24 21
20, 22 21
20, 24 22
22, 24 23
Sample
Mean
19 1 1/6
20 1 1/6
21 2 1/3
22 1 1/6
23 1 1/6
TOTAL 6 1

Sample Means
Solution – sampling without
replacement
4 2
6C 
Sample
Mean
19 1 1/6
20 1 1/6
21 2 1/3
22 1 1/6
23 1 1/6
TOTAL 6 1
X P(X)
19 1/6
20 1/6
21 1/3
22 1/6
23 1/6

Sample Means
Consider a population with
values
1, 2, 3.
a. List all possible samples of size 2
from the population if sampling is
with replacement.
b. Construct a sampling distribution of
the sample means for this
population.

Sample Means
values
2, 5, 7, 8.
without replacement.
population.

Sample Means
values
1, 1, 2, 3, 4.
without replacement.
population.

Check your
understandingA group of students got the following
scores in a test: 6, 9, 12, 15, 18, and 21.
Consider sample scores of size 3 that can
be drawn without replacement from the
given scores.
a. List all the possible samples and the
corresponding means.
b. Construct the sampling distribution of
the sample means.

CABT SHS Statistics & Probability - Sampling Distribution of Means

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CABT SHS Statistics & Probability - Sampling Distribution of Means