Chapter IV Estimation of Parameters Statistics and Probability PDF

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Self-Learning Module for Grade 11MathematicsStatistics and ProbabilityCHAPTER IV: ESTIMATION OF PARAMETERSLESSON 1: Determining Point Estimate of the Population MeanIn some instances, we need to assume certain value that may represent the whole. In assuming a certain value can done by using single n...

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Self-Learning Module for Grade 11 Mathematics Statistics and Probability

CHAPTER IV: ESTIMATION OF PARAMETERS LESSON 1: Determining Point Estimate of the Population Mean Introduction In some instances, we need to assume certain value that may represent the whole. In assuming a certain value can done by using single number or by considering range of values, for a purpose of having higher chance that the TRUE value may be included. In this module, the other two type of estimates, can be illustrated.

ow to Use this Module This module contains pretest, the lesson proper, reflection, and the post test. e starting with the lesson proper, the learner has to answer the pretest. After the t, read and engage with the lesson proper by following or doing the activities, then er the post – test right after.

PRE-TEST Direction: Shade/blacken the circle that corresponds to the correct answer. A B C D

A B C D

A B C D

A B C D

A B C D

1. Which of the following statements is statistically true about point estimate? A. It assumes single number/quantity B. It considers range of data/value C. It has concern and upper found/limit D. It assumes multiple number/quantity 2. Which of the following is considered as point estimate? A. The assume height of the two Story building is 7m to 9m B. The assume height of the building is less than 9m C. The two-storey building estimated to be 8m high D. The estimated height of the two-storey building is more than 8m. 3. An estimate is interval if __________________.? A. It assumes single number/quantity B. It considers range of data/value C. It assume exact value or data D. No Value has determined 4. The following is an example of interval estimate EXCEPT A. The average age of G11 Students is assumed to be from 16 to 18 years. B. The average age of G11 Students is from 18 to 19 years. C. The average age of G11 Students is 17 years. D. The average age of G11 Students is from 15 to 16 years. 5. What must be done to make high chances of estimate for the population mean? A. Apply the point estimation. B. Use the interval estimate. C. Ignore using range of values.

D. Identify a single estimated value.

Competency 1. Illustrates point and interval estimates (M11/SP12-IIIf-2) 2. Distinguishes between point and interval estimations (M11/SP12-IIIf-3)

Objective At the end of this lesson, you should be able to: 1. Illustrate point and interval estimate. 2. Identifies point and interval estimates.

Procedure/Learning Experience Activity Priming Consider the illustration below in reading the texts that follow.

A

B

Activity 1 Reading the texts Albert and Boyet agreed to catch fish using spears. Albert used single spear because it is lightweight and easy to use. Boyet, on the other hand decided to use a multiple spear-ended one. When they are already in the fishing ground, each of them saw fish and aim at it.

Analysis Challenge yourself to think about the questions given based from our activity.

Analysis 1 Answer the following questions based on the previous activity. 1.

How do you differentiate the spears used by the person in the given situation?

2.

Who do you think will hit the target with higher probability? Why?

3.

Which do you think can assume to be the point estimate? The interval estimate? 146

Activity 2 Study the illustrations below. The father and his son are taking the possible height of the building opposite them.

Father, do you agree with me that the building is about 15 meters high?

No son. The building is about 14 meters to 16 meters high.

Analysis 2 1. 2. 3. 4.

What can you say about the estimated height given by the son? How many possible value/s is/are within the estimates of the father? Which do you think is the point estimate? the interval estimate? What can you say about the point estimate? The interval estimate?

Abstraction Check your answers and analysis here.

Point estimation is the process of finding a single value, called point estimate, from a random sample of the population to approximate a population parameter. Interval estimate is a range of values within which the population mean is more likely to be located.

Application Since you already know a lot of things about the topic, let us assess that knowledge!

Using the rectangular block at the right, a.

Make a point estimate on the possible number of cubes it is made of. Answer: ___________________________

b.

What interval estimate would assume the number of unit cubes has the block. Answer : __________________________________

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Reflection What have you learned from this lesson? Write in 2 to 3 sentences, in paragraph form. _____________________________________________________________________________ _____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________

POST - TEST Direction: Shade/blacken the circle that corresponds to the correct answer. A B C D

A B C D

A B C D

A B C D

A B C D

1. Which of the following statements is true about point estimate? A. It assumes single number/quantity B. It considers range of data/value C. It has lower and upper bound/limit D. It assumes multiple number/quantity 2. Which of the following illustrates a point estimate? A. The assume height of the two-storey building is 7m to 9m B. The assume height of the building is less than 9m C. The two-storey building estimated to be 8m high D. The estimated height of the two-storey building is more than 8m 3. Which of the following statements best describe an interval estimate ? A. It targets a single number/quantity B. It assumes range of data/value within which parameter is more likely to be found. C. It assumes exact value or data D. No Value has determined 4. The following does not illustrate interval estimate? A. The average age of G11 students is from 16 to 18 years old B. The average age of G11 students is from 18 to 19 years old C. The average age of G11 students is from 17 years old. D. The average age of G11 students is from 15 to 16 years old 5. If the researcher wants to have higher probability containing the population mean in the estimated value, he must _____. A. Use the point estimate. B. Assume range of values as interval estimate. C. Make only one target value.

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D. Refuse to estimate.

LESSON 2: Parameter and Statistic Introduction Population consists of large set of data that is difficult to deal with, thus representation of the population parameters is necessary. In the representation of the population, random samples are drawn from it. This random sample can be the basis in describing the characteristics of the whole population, as statistics are obtained in this sample. One of the statistics used to describe the population is point estimator, which assumes the possible value of the population mean. In this module, the point estimator from the population mean is discussed.

How to Use this Module This module contains pre-test, the lesson proper, reflection, and the post test. Before starting with the lesson proper, the learner has to answer the pre test. After the pretest, read and engage with the lesson proper by doing the activities, then answer the post test right after.

PRE-TEST Direction: Shade/blacken the circle that corresponds to the correct answer. A B C D

A B C D Sample (5,6) (6,12) (12,16) (16,20)

A B C D A B C D A B C D

1. Which of the following statements tells the correct idea of point estimator? A. The variance is the same as point estimators B. The Standard Deviation is also the point estimators of population mean C. The sample size is also the point estimators of the population mean D. The sample mean is the point estimator for the population mean. X 5.5 9 14 18

Count the following Numbers as population: 5, 6, 12, 16, 20 where mean is 11.8 The table at the left contains samples and their corresponding sample means ( ẍ ). Use the table in answering questions 2 to 5.

2. What is the point estimator for the population mean if the samples are 5 and 6? A. 6.5 B. 9 C. 14 D. 18 3. If we have the samples (12,16) what is it point estimator? A. 5.5 B. 9 C. 14 D. 18 4. The point estimator is 9, which samples are drawn from the population? A. 5 & 6 B. 6 & 12 C. 12 & 16 D. 16 & 20 5. Which of the following is NOT a point estimator from the mean population? A. 5.5 B. 10 C. 14 D. 18

Competency

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1. Identifies point estimator for the population mean. (M11/SP12-IIIf-4)

Objective At the end of this lesson, you should be able to: 1. Draws samples from a given population. 2. Identifies point estimator for the population mean.

Procedure/Learning Experience Activity Priming

Consider the 5 households and the number of residents.

Illustration

A

C

E

6 residents

8 residents

B

D

3 residents

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2 residents

9 residents

What do these pictures above represent? The households show different number of residents. Activity: 1 List down the number of household members from the illustration above. Based from the data, the population mean ( µ ) is 5.6 or 6 - Average number of residents in the households. Activity 2 Draw two samples from the population such as households A and B. where there 6 and 2 residents respectively. Get the sum of 6 and 2, and divide the sum by the number of households. The sample mean that is obtained is 4, then 4 is the point estimate of the population mean. This means that the estimated average household members is 4, not the actual mean. If we draw another two samples from the population such as household A and C, there are 6 and 8 members. The sample mean is 7, then 7 is the point estimate of the population mean. Let us consider that the samples from A ,D and E, which are 6, 9 and 3 are used. What is the total household from the three (3) residences? What is the sample mean? The sample mean of 6 is obtained. This means that the point estimate of the population mean if the samples are 6, 9, and 3 is 6, which happened to be the actual population mean. Moreover, if samples 6 and 3 are drawn from the sample, the point estimator is 4.5 or 5, since we are dealing with number of people. The table below shows samples and the respective means as the results of the previous activities. Sample means ( ẍ ) 4 7 6 4.5 or 5

Sample ( 6, 2 ) ( 6, 8 ) ( 6, 9, 3 ) ( 6, 3 )

Each mean ẍ from each of the sample is a point estimate of the population mean, also the sample means are the point estimators for the population mean.

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Analysis Challenge yourself to think about the questions given based from our activity.

Answer the following questions.

A b Where did you get point estimators? 1. s What do the sample means of point estimators have in common? 2. tr a c ti o n

Check your answers and analysis here as to The sample means are the point estimators for the population mean.

Application Please share your own thoughts. Believe me, you can!

Situation UNGOS NATIONAL HIGH SCHOOL – Lubayat Extension has an average number of students per section of 32, including Senior High School. If the students in two sample sections are 28 and 34, with the sample mean of 31, give the point estimator for the population mean. Answer: ________________________

Reflection Is there any pros and cons in solving for the population parameter rather than the sample statistic? If there is, what are those? If none, why?

What have you learned from this lesson? Write in 2 to 3 sentences in paragraph form.

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_________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________

POST - TEST Direction: Shade/blacken the circle that corresponds to the correct answer. A B C D

A B C D Sample (5,8) (8,12) (12,15) (15,20)

1. Which of the following statements tell the correct idea of point estimator? A. The population variance is the same as point estimators. B. The Standard Deviation is also the point estimators of population mean C. The sample size is also the point estimators of the population mean D. The sample mean is the point estimator for the population mean X 6.5 10 13.5 17.5

Count the following Numbers as population: 5, 8, 12, 15, 20 where mean is 12. The table at the left contains samples and their corresponding sample means ( ẍ ). Use the table in answering questions 2 to 5.

2. What is the population estimator for the population mean if the samples are 5 and 8? A. 6.5 B. 10 C. 13.5 D. 17.5 A B C D

3. If we have the samples (12,15) what is its point estimator? A. 6.5 B. 10 C. 13.5 D.17.5

A B C D

4. The point estimator is 10, which samples are drawn from the population? A. 6.5 B. 10 C. 13.5 D.17.5

A B C D

5. Which of the following is NOT a point estimator for the population mean? A. 6.5 B. 11 C. 13.5 D.17.5

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LESSON 3: Computing Point Estimates of the Population Mean Introduction In some instances, we need to assume certain value that may represent the whole. In assuming a certain value can be done by using single number or by considering range of values, for a purpose of having higher chance that the TRUE value may be included. In this module, the other two types of estimates, can be illustrated.

w to Use this Module This module contains pretest, the lesson proper, reflection, and the post test. Before starting with the lesson proper, the learner has to answer the pretest. After the pretest, read and engage with the lesson proper by following or doing the activities, then answer the post test right after.

PRE-TEST Direction: Shade/blacken the circle that corresponds to the correct answer. Consider the ff. situation answering the post-test. The time used by 5 online gamers a day are 1, 2, 3, 4, 5 hours. A B C D

A B C D

1. How many sample need to be drawn from the population to obtain the point estimator? A. 1 B. At least one C. At most two D. 5 2. If 3 samples are drawn from the population, how do you obtain the point estimator? A. Add the sample size B. Divide the sample size C. Divide the sum by the sample size D. Divide the product by the sample size

A B C D

3. Using the samples 2 and 3, what is the point estimator? A. 2.5 B. 3.5 C. 4.5 D. 5.5

A B C D

4. If the samples are 1, 4, and 5, what is the point estimator? A. 6.3 B. 5.3 C. 4.3 D. 3.3 5. Using the sum 45 derived from a large population of 5 samples, what is the point estimator for the population mean? A. 9 B. 10 C. 11 D. 12

A B C D

Competency 154

1. Computes for the point estimates of the population mean.(M11/SP12-IIIf-5)

Objective At the end of this lesson, you should be able to: 1. Draw/create samples from the given population. 2. Compute from the point estimate of the population mean.

Procedure/Learning Experience Activity In this lesson, you will learn to compute for the point estimate of the population mean. Review of the previous lesson. Compute for the point estimate of the population mean Recall previous

Sample

ẍ

(5,8)

6.5

(8,12)

10

(12,15)

13.5

(15,20)

17.5

that the table at the left contains data from the lesson. Which are the point estimators for the population mean?

Activity: 1 Using the number of residents in each household which are 6, 2, 8, 9, 3, and whose average is 5.6 or 6, let us compute for the point estimator of the population mean if two samples are drawn from the population, such as 6 and 8. The Point Estimator = ( ẍ ) = where x1 and x2 are the samples and n is the sample size ẍ = = = = 7.5 Therefore the point estimator if the samples are 6 and 8 is 7.5 or 8. Activity 2 How about if 3 samples are drawn from the population such as 2, 3, and 9? The point estimator for the population is determined by the solution below. ẍ= ẍ=

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ẍ= ẍ = 4.7 or 5 Therefore the point estimator for the population mean is 5 , since it refers to the number of people. Activity 3 Consider a large population of teenagers whose average age is 17.5. Thirty six (36) samples are drawn from this population. The sum of the ages of the sample teenagers is 594, what is the point estimator for the population mean. Given: Σ x = 594, n = 36 µ = 17.5 Solution ẍ= ẍ= ẍ = 16.5

A n a Therefore the point estimator of the population is 16.5 l y s i s Answer the following questions. 1. 2.

Which is considered as the point estimator for the population mean? How do the point estimator for the population mean be obtained?

Abstraction Check your answers and analysis here.

The sample mean is the point estimator for the population mean. Point estimator is obtained by calculating the sample mean.

Application Let us apply your knowledge in the following data below. Using the given population of teachers in four ( 4 ) departments, such as in Math department is 3, in MAPEH department is 7, in English department is 4, and in Filipino department is 5. Find the point estimator for the population mean

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Challenge yourself to think about the questions given based from our activity.

a) if the sample are 3, 4, and 5,

Answer: ____________________

b) if the sample are 4, 6, and 7.

Answer: _____________________

Reflection What have you learned from this lesson? Write in 2 to 3 sentences in paragraph form.why? ____________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________

POST - TEST Direction: Shade/blacken the circle that corresponds to the correct answer. Consider the given situation in answering the questions that follow. The number of hours the 5 individual rested a day are 5, 6, 7, 8, 9For numbers 1-5, a population consists of the data (1, 2, 3, 4). A B C D

1. How many sample/s is/are needed to be drawn from the population to obtain the point estimator? A. 1 B. At least two C. At most on D. 5

A B C D

2. If 3 samples are drawn from the population, how do you obtain the point estimator? A. Add the sample size B. Divide the sampl...