Statistics Probability Quarter 3 Module 4: Random Sampling, Parameter and Statistic PDF

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Statistics and Probability- Grade 11 Quarter 3 – Module 4: Random Sampling, Parameter and StatisticRANDOM SAMPLINGWhat I KnowChoose the letter of the best answer. Write the chosen letter on a separate sheet of paper. It refers to the entire group that is under study or investigation. A. populationB....

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Statistics and Probability- Grade 11 Quarter 3 – Module 4: Random Sampling, Parameter and Statistic RANDOM SAMPLING What I Know Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. It refers to the entire group that is under study or investigation. A. population B. sample C. random sampling D. subset 2. It is a subset taken from a population, either by random or nonrandom sampling techniques. A. population B. sample C. random sampling D. lottery sampling 3. What sampling technique in which members of the population are listed and samples are selected in intervals called sample intervals. A. systematic sampling B. lottery sampling C. stratified random sampling D. quota sampling 4. It is sometimes called area sampling, it applies on a geographical basis. A. systematic sampling B. lottery sampling C. stratified random sampling D. cluster sampling

5. It refers to a part of the sampling technique where each sample point has an equal chance of being selected. A. systematic sampling B. lottery sampling C. random sampling D. quota sampling Lecture 1: Random Sampling What’s In (Introduction) If a researcher wants to observe, examine or test a theory or hypothesis, he will consider the problem by selecting a section of the population of the study using a method called random sampling. In random sampling, all subjects in the population listed in the study have the same chances of being chosen for the survey. This means that, ultimately, each member of the sample retains characteristics, or impartial characteristics, of the population. With random sampling, the conclusions of the post-hypothesis tests applied to the sample selection will apply to the entire population as well. This is due to the fact that the selection of the sample essentially represents the characteristics of the population from which it is obtained, since each member of the sample was drawn unbiased from the population data. When bias in sample selection is avoided, the results of a particular study are considered more conclusive and the error is minimized.

What’s New (exercise) Analyze the following study. 1. Mrs. Dela Cruz wants to get the analysis on her pre-test in Statistics and probability of grade 11 students in ABC high school with 150 students in the subject. Should she get the scores of one class only?

Analysis: Mrs. Dela Cruz class is not that big, it is much better if she will get the scores of her entire class to have an analysis. What is It

(Terminologies)  The population refers to the whole group under study or investigation.  In research, the population does not always refer to people. It may mean a group containing elements of anything you want to study, such as objects, events, organizations, countries, species, organisms, etc.  A sample is a subset taken from a population, either by random sampling or by non-random sampling. A sample is a representation of the population where it is hoped that valid conclusions will be drawn from the population.

Random sampling is a selection of n elements derived from the N population, which is the subject of an investigation or experiment, where each point of the sample has an equal chance of being selected using the appropriate sampling technique.

Types of Random Sampling Techniques 1. Lottery sampling is a sampling technique in which each member of the population has an equal chance of being selected. An instance of this is when members of the population have their names represented by small pieces of paper that are then randomly mixed together and picked out. In the sample, the members selected will be included. 2. Systematic sampling is a sampling technique in which members of the population are listed and samples are selected at intervals called sample intervals. In this technique, every nth item in the list will be selected from a randomly selected starting point. For example, if we want to draw a 200 sample from a population of 6,000, we can select every 3rd person in the list. In practice, the numbers between 1 and 30 will be chosen randomly to act as the starting point. 3. Stratified random sampling is a sampling procedure in which members of the population are grouped on the basis of their homogeneity. This technique is used when there are a number of distinct subgroups in the population within which full representation is required. The sample is constructed by classifying the population into subpopulations or strata on the basis of certain characteristics of the population, such as age, gender or socio-economic status. The selection of elements is then done separately from within each stratum, usually by random or systematic sampling methods. Example: Using stratified random sampling, select a sample of 400 students from the population which are grouped according to the cities they come from. The table shows the number of students per city. City

Population (N)

A

12,000

B

10,000

C

4,000

D

2,000

Solution: To determine the number of students to be taken as sample from each city, we divide the number of students per city by total population (N= 28,000) multiply the result by the total sample size (n= 400). City A

Population (N) 12,000

B

10,000

C

4,000

D

2,000

Sample (n) 12,00 0 x 28,00 0 10,000 28,000

(400)= 171

x (400)= 143

4,000 x (400)= 57 28,000 2,000 x (400)= 29 28,000

4. Cluster sampling is sometimes referred to as area sampling and applied on a geographical basis. Generally, first sampling is performed at higher levels before going down to lower levels. For example, samples are taken randomly from the provinces first, followed by cities, municipalities or barangays, and then from households. 5. Multi-stage sampling uses a combination of different sampling techniques. For example, when selecting respondents for a national election survey, we can use the lottery method first for regions and cities. We can then use stratified sampling to determine the number of respondents from selected areas and clusters.

What’s More On your answer sheet, give one situation where each of the sampling methods is being applied. 1.

Lottery Sampling: ___________________________________________________________ ________________ ___________________________________________________________ ________________ ____________________________________

2.

Systematic Sampling: ___________________________________________________________ ________________ ___________________________________________________________ ________________ ____________________________________

3.

Stratified Random Sampling ___________________________________________________________ ________________ ___________________________________________________________ ________________ ____________________________________

4.

Cluster Sampling ___________________________________________________________ ________________ ___________________________________________________________ ________________ ____________________________________

5.

Multi-stage sampling ___________________________________________________________ ________________ ___________________________________________________________ ________________ ____________________________________

What I Have Learned Identify the terms being described and write your answer on a separate sheet of paper. 1. It refers to the entire group that is under study or investigation. 2. It is a subset taken from a population, either by random or non-random sampling technique. A sample is a representation of the population where one hopes to draw valid conclusions from about population. 3. This is a selection of n elements derived from a population N, which is the subject of the investigation or experiment, where each sample point has an equal chance of being selected using the appropriate sampling technique. 4. A sampling technique where every member of the population has an equal chance of being selected. 5. It refers to a sampling technique in which members of the population are listed and samples are selected in intervals called sample intervals. What I Can Do If you were a researcher and wanted to conduct a research within your Barangay, what would it be? What sampling technique are you going to use? Assessment A. Identify the type of sampling method. Write your answer on a separate sheet of paper. __________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 4 th 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 raffle box.

Additional Activities Get the samples needed for each category using stratified random sampling. There are 20 members of taekwondo club, 40 math club members, 60 drama theatre members, and 30 members of science club. The researchers want to get 20 respondents out of these organizations. Identify the samples to be taken in each organization.

Lecture 2: Parameter and Statistic What I Know Determine the statement whether it is true or false. Write T if the statement is true and F if it is false. Write your answer on a separate sheet of paper. _____1. A statistic is a number which describes a sample. _____2. A parameter is a descriptive measure of population. _____3. An example of parameter is the sample mean. _____4. The value of a parameter can be approximated and is not necessarily equal to the statistic of a sample. _____5. An example of statistic is a population mean.

What’s In  In this course, the parameters and statistics are closely related terms that are important for the determination of the sample size.  Many have trouble understanding the difference between the parameter and the statistic, but it's important to know exactly what these measures mean and how to distinguish them. What’s New

Study the cases below. Identify which of the cases involves measures from a population and a sample. 1. A researcher randomly selected a sample of 1000 people in Barangay, 143 and asked if they used a certain coffee product and 40% of them said yes. 2. A researcher interviewed all the students in a certain school to identify their insights about their favourite shoe brand. Analysis: In the first case the researcher measures for a sample. Only 40% out of 100 said yes. While in the second case the researcher measures the population because the researcher interviewed all the students of that school.

What is It

 A parameter is a descriptive population measure. It is a measure of the characteristics of the entire population (a mass of all the units under consideration that share common characteristics) based on all the elements within that population. Example:

1. All people living in one city, all-male teenagers worldwide, all elements in a shopping cart, and all students in a classroom. 2. The researcher interviewed all the students of a school for their favorite apparel brand. Statistic is the number that describes the sample. It can be calculated and observed directly. The statistic is a characteristic of a population or sample group. You will get the sample statistic when you collect the sample and calculate the standard deviation and the mean. You can use sample statistic to draw certain conclusions about the entire population. Example: 1.Fifty percent of people living in the U.S. agree with the latest health care proposal. Researchers can’t ask hundreds of millions of people if they agree, so they take samples or part of the population and calculate the rest. 2.Researcher interviewed the 70% of covid-19 survivors.

What’s More Give 5 examples of parameter and 5 examples of statistic. Write your answer on a separate sheet of paper.

What I Have Learned To generalize your learned skills and concepts, take note of the similarities and differences of parameter and statistic. Direction: On your answer sheets, draw a Venn diagram by listing the similarities and differences of parameter and statistic.

What I Can Do In conducting a research, which measure are you going to use? Parameter or statistic? Why? ________________________________________________________________ _____ ________________________________________________________________ ___________

Assessment Decide whether the statement describes a parameter or statistic. Write your answer on a separate sheet of paper. 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.

Additional Activities Give a situation in your area that is an example of parameter and statistic. Explain why you considered it as a parameter or a statistic. ___________________________________________________________ __________ ________________________________________________________________ ___________ ____________________________________________________

Lecture 3: Sampling Distribution of the Sample Means What I Know

Round off your answers to two decimal places.

1. 2. 3. 4. 5.

4, 12, 34, 45, 6 23, 45, 67, 89, 21, 11 88, 87, 86, 89, 88, 90 34, 21, 45, 67, 23 12, 9, 6, 5, 32, 40

What’s In In the previous lesson, you have learned the concept about the parameter and statistic. In this lesson we will study a form of probability distribution which is known as the sampling distribution.

What’s New 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

What is It 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 5C2 = 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 2, 4 3 2, 6 4 2, 10 6 2, 12 7 3, 10 6.5 3, 6 4.5 3, 12 7.5 6, 10 8 6, 12 9

Observe that the means vary from sample to sample. Thus, any mean based on the sample drawn from a population is expected to assume different values for samples.

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 or 0.1 3 or 0.1 4 or 0.1 4.5 or 0.1 6 or 0.1 6.5 or 0.1 7 or 0.1 7.5 or 0.1 8 or 0.1 9 or 0.1 1 Observe that all sample means appeared only one; thus, their probability is P(x)= 10 or 0.1

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.

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? To answer this, use the formula NCn, 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= 3 5C3 = 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 86, 88, 90 88 86, 90, 95 90 86, 90, 98 91 86, 90, 95 90 86, 90, 98 91 86, 95, 98 93 88, 90, 95 91 88, 90, 98 92 88, 95, 98 94 90, 95, 98 94 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 or 0.1 90 or 0.2 91 or 0.3

92 93 94

or 0.1 or 0.1 or 0.2

1 Observe that 88, 92 and 93 appeared only once; thus their probability is P(x)= or 10 0.1. Since 90 and 94 appeared twice, their probability is P(x)= or 0.2. While 91 3 appeared thrice, their probability is P(x)= or 0.3 10 Observe that the total probability of all sample means must be equal to

1.

What’s More A population consists of the numbers 2, 4, 8, 10 and 5. Let us list all the possible samples of size 3 from this population and construct the sampling distribution of the sample mean.

What I Have Learned Complete the statement by filling in the blank. Write your answer on a separate sheet of paper. A ___________ is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population. To get the possible samples use the formula ______, where N is the ________ and n is the ____ size to be

taken. The total probability of the sample mean must be equal to ____.

What I Can Do Construct a sampling distribution of sample mean and answer the questions on your answer sheet. Samples of 3 cards are drawn from a population of five cards numbered from 1-5. 1.How many are the possible outcomes? 2.What are the possible means? 3.What is the probability of getting 4 as a mean? 4.What is the probability of getting 2 as a mean? 5.What is the probability of getting 3.33 as a mean?

Assessment

Construct all random samples consisting three observations from the given data. Arrange the observations replacement and repetition.

86

89

92

95

Additional Activities

98

in

ascending

order

without

Construct all random samples consisting two observations from the given data. You are asked to guess the average weight of the six watermelons by taking a random sample without replacement from the population.

Watermelon

A

B

Weight (in pounds)

19

14

C 15

D

E 9

10

F 17

Lesson 3

Lesson 3

What I have learned 1 Sample distribution of . sample mean 2 NCn 3 . Population . Sample 4 . 1 5 .

What’s more: Sample mean 3.61 7 1 4.6 75 1 5.31 3 5.6 7 6.31 3 1 6.6 7 1 7.3 3 1 7.6 7

Lesson 3

P(x) /10 or 0.1 /10 or 0.1 /10 or 0.1 /10 or 0.1 2/10 or 2 0. /10 or 0.1 /10 or .33 0.1 /10 or 0.1 /10 or 0.1

What I Know: 1. 20.2 2. 3 . 42.67 88 4 . 38 5. 17.33 What I can 1 . do: 10 2 . 2, 2 ,,2.67,3,3.33,3.67, 4 3 . 1/ 10 4. 0.1 5. 0.2

Lesson 2

Lesson 2

Lesson 2

Lesson 2

Assessment

Additional Activities :

What’s more:

What I know:

Answer may vary.

Answer may vary.

1. 2. 3. 4. 5.

Statistic Parameter Parameter Statistic parameter

WhatI Have learned: Answer may vary.

1 .T 2 .T

What I can do:

3 .F

Answer may vary

4 .T 5 .F

Lesson 1

Lesson 1

Lesson 1

Lesson 1

Additional Activities:

Assessment :

What I can do:

W...