cabt shs statistics & probability - sampling distribution of means

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CABT Statistics & Probability – Grade 11 Lecture Presentation

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Sampling Distribution of

MeansA 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 votersCell 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 testingRandom 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:

21NnNe

Example: The number of samples needed for a population of 1,000 at a margin of error of 5% is

21000 286

1 1000 0.05n

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

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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

Nkn

Example:To get a sample containing 8 students from a group of 40 students using systematic random sampling, the sampling interval is40 58k



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

PopulationStrata

Number of Students per Strata

No. of Samples per Strata

Male 340 20

Female 500 30

TOTAL 840 50

340 50 20.24 20840males

500 50 29.76 30840females

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

Do you have any QUESTIONs?

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).




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Parameter or statistic?1. All CABT Senior High School

students2. A selected group of CABT Senior

High School students3. Average scores of Grade 10 NAT

takers nationwide4. 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 understanding1. 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 PopulationSTEP 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 distribution for the sample means X.




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

nNN nC




1Consider 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.




1Solution – sampling with replacementNumber 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

24 16




1Sample Mean Frequency Probability

18 1 1/1619 2 1/820 3 3/1621 4 1/422 3 3/1623 2 1/824 1 1/16

TOTAL 16 1

X P(X)18 1/1619 1/820 3/1621 1/422 3/1623 1/824 1/16

Solution – sampling with replacementNumber of samples of size 2:

24 16




1Solution – sampling without replacementNumber of samples of size 2:4 2 6C

Sample Sample Mean

18, 20 1918, 22 2018, 24 2120, 22 2120, 24 2222, 24 23

Sample Mean Frequency Probability

19 1 1/620 1 1/621 2 1/322 1 1/623 1 1/6

TOTAL 6 1




1Solution – sampling without replacementNumber of samples of size 2:

4 2 6C

Sample Mean Frequency Probability

19 1 1/620 1 1/621 2 1/322 1 1/623 1 1/6

TOTAL 6 1

X P(X)

19 1/620 1/621 1/322 1/623 1/6




2Consider a population with values1, 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.




3Consider a population with values2, 5, 7, 8.a. List all possible samples of size

2 from the population if sampling is without replacement.





4Consider a population with values1, 1, 2, 3, 4.a. List all possible samples of size

3 from the population if sampling is without replacement.




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.

Do you have any QUESTIONs?

Summing it up!

Thank you!

cabt shs statistics & probability - sampling distribution of means

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