Statistics Probability Q3 Mod4 Random Sampling, Parameter and Statistic PDF

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Statistics andProbabilityQuarter 3 – Module 4:Random Sampling,Parameterand Statistic, and SamplingDistribution of StatisticsStatistics and Probability- Grade 11Alternative Delivery ModeQuarter 3 – Module 4: Random Sampling, Parameter and Statistic, andSampling distribution of statisticsFirst Edition...

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Statistics and Probability Quarter 3 – Module 4: Random Sampling,Parameter and Statistic, and Sampling Distribution of Statistics

Statistics and Probability- Grade 11 Alternative Delivery Mode Quarter 3 – Module 4: Random Sampling, Parameter and Statistic, and Sampling distribution of statistics First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

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Statistics and Probability Quarter 3 – Module 4: Random Sampling, Parameter and Statistic, and Sampling Distribution of Statistics

Introductory Message This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson. Each SLM is composed of different parts. Each part shall guide you step-bystep as you discover and understand the lesson prepared for you. Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to selfcheck your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these. In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning. Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task. If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Thank you.

What I Need to Know

This module was designed and written with you in mind. It is here to help you master the Statistics and Probability. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. The module is divided into three lessons, namely:   

Lesson 1 – Random Sampling Lesson 2 – Parameter and Statistic Lesson 3 – Sampling Distribution of statistics (sample mean)

After going through this module, you are expected to: 1. illustrate random sampling; (M11/12SP-IIId-2) 2. distinguish between parameter and statistic; (M11/12SP-IIId-3) and 3. identify sampling distribution of statistics (sample mean).(M11/12SP-IIId-4)

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Lesson

1

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

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Lesson

1

Random Sampling

What’s In 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.

Notes to the Teacher The students should understand the importance of having a random sampling in research.

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What’s New 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 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.

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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 A B C D

Population (N) 12,000 10,000 4,000 2,000

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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) 12,000 28,000 10,000 28,000 4,000 28,000 2,000 28,000

x (400)= 171 x (400)= 143 x (400)= 57 x (400)= 29

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: ___________________________________________________________________________ ___________________________________________________________________________ ____________________________________

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3.

Stratified Random Sampling ___________________________________________________________________________ ___________________________________________________________________________ ____________________________________

4.

Cluster Sampling ___________________________________________________________________________ ___________________________________________________________________________ ____________________________________

5.

Multi-stage sampling ___________________________________________________________________________ ___________________________________________________________________________ ____________________________________

What I Have Learned

1. 2.

3.

4. 5.

Identify the terms being described and write your answer on a separate sheet of paper. It refers to the entire group that is under study or investigation. 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. 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. A sampling technique where every member of the population has an equal chance of being selected. 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?

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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 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 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.

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Lesson

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.

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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

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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

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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.

_____________________________________________________________________ ___________________________________________________________________________ ____________________________________________________

Lesson Sampling Distribution of the

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Sample Means

What I Know Given the set of numbers, compute for the mean. Write your answer on the space provided in each item. 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

12

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 2,3

mean 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 poss...