Statprb Quarter 3 Module 5 ( Final) PDF

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Department of Education SCHOOLS DIVISION OF LAOAG CITY Laoag City66WHOLE BRAIN LEARNING SYSTEMOUTCOME-BASED EDUCATIONLEARNINGMODULESTATISTICS ANDPROBABILITYQUARTER3WEEK 5####### GRADE####### 11MODULE INSTATISTICS AND PROBABILITYQUARTER 3WEEK 5####### Sampling and Sampling DistributionDevelopment Tea...

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66

WHOLE BRAIN LEARNING SYSTEM OUTCOME-BASED EDUCATION GRADE

STATISTICS AND

11

PROBABILITY Department of Education

LAOAG SCHOOLS DIVISION OF LAOA G CITY Laoag City

LEARNING MODULE

QUARTER WEEK

3 5

MODULE IN STATISTICS AND PROBABILITY

QUARTER 3 WEEK 5

Sampling and Sampling Distribution Development Team Writers:

Jerick S. Paltong

Michael G. Calipjo

Editors/Reviewers: Gerson Jeremy C. Antonio Illustrator:

Jeshimon C. Patoc

Layout Artist:

Vanessa Miguel

Myla Fei Martinez

Ma. Teresa R. Pascual Gregorio C. Agatep, Jr.

Management Team: Vilma D. Eda

Arnel S. Bandiola

Lourdes B. Arucan

Juanito V. Labao

Marlyn S. Ventura

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What I Need to Know In this module, we shall learn how to construct the sampling distribution of the sample means. In the succeeding lessons, we shall find out some important characteristics of the sample distribution of the sample means. This will eventually help us understand the process of making statistical inferences about the population using a sample drawn from it. Most Essential Learning Competencies (MELCs) The learner: 1. illustrates random sampling; 2. distinguishes between parameter and statistic; and 3. identifies sampling distributions of statistics (sample mean). OBJECTIVE: At the end of the module, the students are expected to: 1. illustrate random sampling; 2. distinguish between parameter and statistic; 3. construct a sampling distribution of sample means; and 4. construct a histogram of the sampling distribution of the sample means.

What I Know Directions: Read the questions carefully. Choose the correct answer from among the choices. Use a separate sheet of paper as your answer sheet. 1. How many samples are possible when a population consists of 4 values with a sample size of 2? A. 2 B. 4 C. 6 D. 8 For nos. 2 and 3, The following are the heights of five students in centimeters. Suppose samples of size 3 are taken from this population of five students. Student Height (in cm) Marie 120 Ma. Consuelo 130 Marjorie 110 Mark 125 Mhanuel 115 2. How many samples are possible? A. 10 B. 8

C. 6

D. 4

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3. What is the mean of the population? A. 110 B. 115 C. 120 4. Which of the following statements is/are correct? A. The mean is greater than the population mean. B. The mean is less than the population mean. C. The mean is equal to the population mean. D. None of the above

D. 125

5. Which of the following statements best describes the relationship between a parameter and a statistic? A. A parameter has a sampling distribution with the statistic as its mean. B. A parameter has a sampling distribution that can determine what values the statistic is likely to have in repeated samples. C. A parameter is used to estimate a statistic. D. A statistic is used to estimate a parameter. 6. Samples of three cards are drawn at random from a population of eight cards numbered 1 to 8. How many possible samples can be drawn? A. 46 B. 56 C. 66 D. 76 For nos. 7 and 8, refer to the situation below. A survey will be given to 100 randomly selected Grade 11 students at Ilocos Norte College of Arts and Trades. 7. What is the population? A. the 100 selected students B. all Grade 11 students C. all Grade 11 students at Ilocos Norte College of Arts and Traders D. all students at Ilocos Norte College of Arts and Trades 8. What is the sample? A. the 100 selected students B. all Grade 11 students C. all Grade 11 students at Ilocos Norte College of Arts and Trades D. all students at Ilocos Norte College of Arts and Trades 9. “A survey about timekeeping might divide the population by time zone, then take 100 random samples per zone”, what type of random sampling will fall? A. Simple Random Sampling B. Systematic Sampling C. Stratified Sampling D. Cluster Sampling 10. A pharmaceutical company wants to test the effectiveness of a new drug. Volunteers are assigned randomly to one of two groups. The first group will receive the new drug; the second group will receive a placebo. Which type of sampling technique to be used in this given situation? A. Simple Random Sampling B. Sampling C. Stratified Sampling D. Cluster Sampling 4

Lesson

RANDOM SAMPLING

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What’s In ACTIVITY 1. Random Hunt Description: This activity focuses on the different types of random sampling in the form of a crossword puzzle. You should find the different types of random sampling from the puzzle. N M N G I J U Y H U O L P

A N D O M S S A A T G B F

A S N O W B A L L V N A B

E T E F G H J Y U I O P I

S R A S Y S T E M A T I C

C D E R S A E R R T Y U Q

T R A T I F I E D N O O U

A B C D E E E C S E C B O

A B S I M P L E A A P B T

B E A S T S E R E R T T A

Random Simple Quota Stratified Cluster Systematic

What’s New A researcher prefers to achieve unbiased results in his or her study. One of the best ways to fulfill this is through the use of random sampling.

What is It TYPES OF RANDOM SAMPLING 1. Simple Random Sampling is a sampling technique in which each member of the sample is selected by the equivalent drawing of lots.

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Example 1: A researcher wants to study the effects of social media on Grade 11 students in Ilocos Norte College of Arts and Trades. He wishes to use the simple random sampling technique in choosing the members of his sample. If there are 1000 Grade 11 students in the school, how many students should be in his sample? Discuss the steps he must take if he wishes to use the lottery method. Solution: Step 1. Determine the number of students that should be in the sample. Use The Slovin’s formula as follows: 𝑁

𝑛 =1+𝑁𝑒 2 where,

𝑛 = number of samples needed 𝑁= population size 𝑒 = margin of error For the margin of error, use 5% or 0.05. Therefore, 1000

𝑛 = 1+(1,000)(0.05)2 𝑛 = 285.7 𝑜𝑟 286 Step 2. Assign a number to each member of the population. In this problem, assign a number to each of the 1,000 students. Step 3. Write the numbers on pieces of paper with the same size and shape. Fold the pieces of paper. Step 4. Put all the folded pieces of paper in a bowl or box. Step 5. Without looking, randomly pick out 286 folded pieces from the bowl or box.

2. Systematic sampling is a random sampling technique in which every kth element of the population is selected until the desired number of elements in the sample is obtained. The value of k is calculated by dividing the number of elements in the population by the number of elements in the desired sample. The value of k is the sampling interval. 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑁 𝑘= 𝑛 where 𝑘 = sampling interval 𝑁 = population size 𝑛 = sample size 𝑘=

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Example 2: In a group of 250 students, how will you select a sample containing 71 students using the systematic sampling technique? Solution: Step 1. Prepare a sampling frame by randomly arranging the 250 students. Step 2. Assign each student a number from 1 to 250. Step 3. Find the sampling interval k. Divide the population size 250 by the sample size 71. 𝑁 𝑘= 𝑛 250 71 = 3.52 𝑜𝑟 4 Step 4. Select any whole number between 0 and k + 1 by simple random technique. The numbers between 0 and k + 1 are 1, 2, 3, and 4. This chosen value is called the random start. 𝑘=

Step 5. Assume that the randomly selected number is 2. Use 2 as the starting number. Step 6. Select every 4th student from the sampling frame starting from the 2nd student. 1,

2,

3,

4,

5,

6,

7,

8,

9,

10,

11,

12,

1st 2nd 3rd The numbers of the sample will then 2, 6, 10, 14, 18, …

13,

14…….. 4th

3. Stratified sampling is a random sampling technique in which the population is first divided into strata, and then samples are randomly separately from each stratum. In stratified sampling, the population is partitioned into several subgroups called strata, based on some characteristics like year, level, gender, age, ethnicity, etc. Example 3: You want to interview 200 students in your school to determine their opinion on the new school uniform. How will you choose your sample by using stratified sampling if 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? Solution: Subdivide the population into several strata. In this problem, subdivide the population into year levels. Then, make a table similar to the following.

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Population N= 6,000 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 Total

Number of Students Per Stratum 1,200 1,100 1,050 940 900 810 6,000

Sample n = 200 40 37 35 31 30 27 200

To obtain the sample size per year level, divide the total number of students per year level by the total number of students in school, and then multiply the result by 200. Then, select the members of each sample by using simple random sampling. For instance, in Grade 7, select the 40 students from the 1,200 students by using SRS. To compute the sample size in each grade level: a. Grade 7: b. Grade c. Grade

1,200

𝑥 200 = 40

6,000 1,050 9: 6,000 𝑥 200 = 35 900 11: 6,000 𝑥 200 = 30

d. Grade 8:

1,100 𝑥 6,000 940

e. Grade 10: f. Grade 12:

200 = 36.667 𝑜𝑟 37

𝑥 200 = 31.33 𝑜𝑟 33

6,000 810 𝑥 6,000

200 = 27

Sometimes, the computation will result in one less than the value of n. If this happens, round up one of the data to the next integer. In this problem, n= 200. If the sum of all the samples per year level is 199 instead of 200, then round up one of the data, which is not a whole number, to the next integer. Cluster Sampling is a random sampling technique in which the entire population is broken into small groups or clusters, and from the selected clusters, random samples will be selected. The data from the randomly selected clusters are the ones that are analyzed. The difference between cluster sampling and stratified sampling is that the sample consists of elements from the selected clusters only. In contrast, in stratified sampling, the sample consists of elements from all the strata. Example 4: A researcher wants to determine who among the families in a small town are using the new detergent product. How is she going to do this using the cluster sampling technique? Solution Step 1. Divide the population into clusters. Use barrios as clusters Step 2. Not all the barrios of the town will be included in the sample. Choose the final barrios by using either simple random sampling or a systematic sampling technique. Step 3. Not all the families in each selected barrio will be included in the study. Select the final families to be included in the sample by using either a simple random sampling or systematic random sampling technique.

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What’s More ACTIVITY 2. Know Me More! Directions: Identify the type of sampling technique used by the researcher in each of the following situations. 1. A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group aged 11 to 12, and three players from a group of boys aged 13 to 14 to form a recreational soccer team. 2. The teacher gave the researcher a list of 125 Grade 11 students. The researcher selected every 15th name on the list. 3. A pollster interviews all human resource personnel in five different high tech companies. 4. A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers. 5. A researcher interviewed people from each town in the province of Ilocos Norte for his research on technologies. 6. A researcher surveyed all anemic patients in each of the four randomly selected hospitals in Ilocos Norte. 7. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital. 8. A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers. 9. A teacher asked her students to fall in line. He instructed one of them to select every 4th student in the line. 10. A researcher selected a sample of n = 100 from a population of 650 by using the Table of Random Numbers. 11. A researcher studying the effects of educational attainment on promotion conducted a survey of 25 randomly selected workers from each of these categories: college graduate, master’s degree, and doctoral degree. 12. A researcher wants to determine who among the families in a small town divided into barrios are using the new HUAWEI cellphone. These are the final barrios to respond in the research. 13. In recent research conducted in a public school, the subjects of the study were selected using the Table of Random Numbers. 14. A researcher is studying the reaction of the students to the ban of soft drinks in the canteen. He interviews every 8th student entering the gate of the school. 15. A teacher conducted a study in her school to determine who was better in English: the girls or the boys.

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What’s More ACTIVITY 3. Let Me Do It Solve the following problems. 1. A researcher wants to study the effects of social media on Grade 11 students in Ilocos Norte College of Arts and Trades. If there are 1,000 Grade 11 students in the school, how many students should there be in his sample? 2. What would be the sampling interval in a group of 250 students with a sample containing 71 students? 3. You want to interview 300 students in your school to determine their opinion on the new school uniform. How will you choose your sample using stratified sampling if there are 1500 students in Grade 7; 1,200 in Grade 8; 1020 in Grade 9; 980 in Grade 10; 1250 in Grade 11; and 2050 in Grade 12? Complete the table below. Population N = 8000 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 Total

Number of Students Per Stratum

Sample n = 300

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Lesson

2

PARAMETER AND STATISTIC

What’s In ACTIVITY 1: Population or Sample? Give it a try: Directions: For the following problems, decide if the situation deals with a “population” data set or a “sample” data set. Explain your decision. 1. Mrs. Pascual wants to do a statistical analysis of students’ final examination scores in her math class for the past year. Should she consider her data to be a population data set or a sample data set? 2. A group of students surveys 100 students from their freshman class to determine the number of pets in each student’s household. The group plans to compute statistical findings on their data and generalize these findings to the homes of all freshmen students. Should the group consider their data to be a population data set or a sample data set?

What’s New This lesson will discuss parameter and statistic A parameter is a measure that describes a population. Parameters are usually denoted by Greek letters. Population mean 𝜇 , population variance 𝜎 2 , and population standard deviation 𝜎 are examples of parameters. On the other hand, a statistic is a measure that describes a sample mean 𝑥, sample variance 𝑠2 , and sample standard deviation 𝑠.

What is It What is the difference between a statistic and a parameter? A statistic and a parameter are very similar. The difference between a statistic and a parameter is that a statistic describes a sample, while a parameter describes an entire population.

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Parameter. It is a measure that describes a population. Statistic. It is a measure that describes a sample. We want to know about these

We have these to work with

Random Selection

Population

Sample Inference

Parameter (𝜇) (population mean)

Statistic ( 𝑥) (sample mean)

Figure 1. Illustration of the relationship between samples and populations. Example 1: The students want to know the average length of a butterfly. Answer: This is a parameter because it is states something about the entire population of butterflies. Example 2: Identify the parameter and statistic in the given situation A researcher wants to estimate the average height of women aged 20 years or older. From a simple random sample of 45 women, the researcher obtains a sample mean height of 63.9 inches. Answer: The parameter is the average height of all women aged 20 years or older. The statistic is the average height of 63.9 inches from the sample of 45 women.

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Example 3: Identify the parameter and statistic in the given situation. A nutritionist wants to estimate the mean amount of sodium consumed by children under the age of 10. From a random sample of 75 children under the age of 10, the nutritionist obtains a sample mean of 2993 milligrams of sodium consumed. Answer: The parameter is the mean amount of sodium consumed by children under the age of ten. The statistic is the mean of 2993 milligrams of sodium obtained from the sample of 75 children.

What’s More ACTIVITY 2. Identify Me!!! Directions: Tell whether the given value is a statistic or a parameter. 1. Jupiter has 67 confirmed moons. 2. The Philippines consists of 7,641 islands. 3. Fifty students who were randomly selected will participate in a study on the effects of using calculators in learning mathematics. 4. In a study concerning employees’ economic status in a certain company, a researcher randomly selected 45 employees. 5. Based on a sample of 900 elementary students, it was found out that 30% of them could not do long division. 6. A teacher randomly selected 15 students from her class. These 15 students were asked to go to the auditorium to be interviewed by a researcher. 7. The Mathematics department consists of 19 male teachers, 12 female teachers, and one department head. 8. A researcher randomly selected twenty-five people. They will take part in a research study on cleanliness. 9. The City of Manila is politically divided into six legislative districts. 10. The Statistics teacher randomly selected 35 questions from previous questions to be given in the coming second periodic examination. 11. It was found out that ten teachers in a certain private high school have doctorate degrees. 12. In a survey of 1,500 bulbs manufactured by a company, it was found out that 120 of them were defective. 13. The Philippine Senate is composed of 24 senators. 14. Based on a sample of 1,200 surveyed students, it was found out that 20% of them needed financial aids. 15. The manager randomly selected 35 employees to participate in the Sports Fest.

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What I Can Do ACTIVITY 3. Give It !!! Directions: Identify the parameter and statistic of the following situations. 1. Nexium is a drug that can be used to reduce the acid produced by the body and heal damage to the esophagus. A researcher wants to estimate the proportion of patients taking Nexium that are healed within eight weeks. A random sample of 224 patients suffering from acid reflux is obtained, and 213 of those patients were healed after eight weeks. Parameter:____________________________________________ Statistics:_____________________________________________ 2. An education official wants to estimate the proportion of adults aged 18 or older who had read at least one book during the previous year. A random sample of 1006 adults aged 18 or older is obtained, and 835 of those adults had read at least one book during the previous year. Parameter:____________________________________________ Statistic:______________________________________________ 3. A school Administrator wants to estimate the ...