Statistics Probability Q3 Mod5 Finding the Mean and Variance PDF

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Statistics and ProbabilityQuarter 3 – Module 5:Finding the Mean and theVariance of the SamplingDistribution of the Sample MeansStatistics and Probability Alternative Delivery Mode Quarter 3 – Module 5: Finding the Mean and the Variance of the Sampling Distribution of the Sample Means First Edition, ...

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Statistics and Probability Quarter 3 – Module 5: Finding the Mean and the Variance of the Sampling Distribution of the Sample Means

Statistics and Probability Alternative Delivery Mode Quarter 3 – Module 5: Finding the Mean and the Variance of the Sampling Distribution of the Sample Means 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. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

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Statistics and Probability Quarter 3 – Module 5: Finding the Mean and the Variance of the Sampling Distribution of the Sample Means

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 self-check 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.

General Note to the Teacher This module contains helpful procedures or strategies that will help you in guiding the learners towards the attainment of the objectives intended for this lesson.

What I Need to Know After successfully completing this self-learning module, you are expected to master essential knowledge and skills about finding the mean and the variance of the sampling distribution of the sample means. Specifically, you are more likely able to: 1. find the mean and variance of the sampling distribution of the sample mean (M11/12SP-IIId-5); and 2. define the sampling distribution of the sample mean for normal population when the variance is: (a) known; (b) unknown (M11/12SP-IIIe-1). These most essential learning competencies will be condensed into a simplified user lesson that will be discussed along the path of your academic journey with this self-learning module. Enjoy your steps towards the attainment of our objectives. Are you ready? If you are, let’s go!

What I Know After you get acquainted with the objectives, it's important to assess your knowledge and experience about the lessons you are just about to discover. First, let us try the next Alternate Response Diagnostic Test Type to check something you already know about this lesson. Consider that your score throughout this part of the module does not actually affect your performance. So, it's all right to get bad scores. Let's get a warm-up! Direction: On your answer sheets, write “WE HEAL” if the statement is correct and write “WE RECOVER” if the statement is incorrect. 1.

2.

Theoretically, a histogram demonstrating the mean of all samples analyzed from a given population would be known as the sampling distribution. The review of sampling distributions of the sample means will be the baseline for inferential statistics and the test hypothesis. 1

3.

The total value of all possible samples means that all possible random samples of a given population size will be equal to the population mean only if the data were normally distributed. 4. For any population, the standard deviation of the sample means is approximately equal to the standard deviation of the population. 5. When the average of all possible values of the sample statistic is equal to the parameter, the statistic is known as a biased estimator of the parameter. 6. The mean of the sampling distribution of the sample means is equal to the mean of the population mean. 7. The standard deviation of the sampling distribution of the mean is also called as the standard error of the mean. 8. The error of sampling calculated from a small sample will always be greater than one calculated from a large sample. 9. Sampling error is the difference between the sample mean and the population mean. 10. To reduce the potential for extreme sampling error, the size of the sample should be reduced.

You can now verify your answers whether they are right or not. How's the result going? Is that good or bad? Don't worry if you have bad scores, because that would be a reliable indicator that you're going to need this self-learning module. That implies there are some sources of competitive advantage that you need to develop and start exploring. So, let's just go! Let us move ahead to the next part of this module so that you can achieve the skills and competencies that you need to improve.

Lesson

1

Finding the Mean and the Variance of the Sampling Distribution of the Sample Means

Recognize that the sampling distribution of the sample means is definitely the probability distribution of the sample means, which also implies that the sample is the random variable in this probability distribution. But since the sampling distribution of the means is the probability distribution of the random variable X, we could perhaps calculate its mean and variance. 2

In this lesson, we shall solve for the mean in the variance of the sampling distribution of the sample means and investigate some of its important properties along with the definition for normal population when the variance is known and unknown.

What’s In In this portion, you will connect your learned concepts and skills from previous lessons, which have much to do with the introduction to sampling distribution of the sample means, to this current lesson, which is finding mean and variance of the sampling distribution of the sample means. There are several concepts from the previous lesson that are directly related to this lesson. Now, the activity given below will help you link those concepts and ideas as you explore the lesson of this module. Direction: With the given jumbled letters, complete the statements below by arranging the letters below. Write your answers on your answer sheets. 1. A – S – N – L – M – I – P – G The _____________ distribution of sample means is frequency distribution of the sample means taken from a population. 2. O – T – M – A – R – S – I – G – H The _____________ of the sampling distribution of the sample means is a bar graph constructed by plotting the sample means along the horizontal axis and the problem along the vertical axis. 3. A – M – E – N The _____________ of the sampling distribution of the sample means is equal to the mean of the population. 4. R – O – R – E – R Standard _____________ of the mean is the standard deviation of the sampling distribution of the sample mean. 5. D – E – X – E – P – E – T – C Mean is also called _____________ value.

Check your answers at this point if they are correct. If this is not the case, you may read the given sentences and analyze why it is completed as it was.

3

Recall the terms discussed in the previous modules and try to link them to the new terms you encountered in this section of the module. For a maximum of 5 minutes, reflect on their similarities and differences so that you can get through this module smoothly. Since you're done reflecting and familiarizing yourself with the terms related to the previous lesson, you can proceed to the next part of this module! Way to go there!

What’s New Now that you have connected the previous lesson to your approach to learning concepts and skills through this module, let us consider the following activity that will give you a recall of the concepts of the previous lesson. Situation: It is Monday! Harvey is very enthusiastic and challenged for his Modular Distance Learning (MDL) experience at Week 5 in Statistics and Probability. His encounter with past quarters and modules justifies his excitement in learning more about the core subject. He chose the MDL's Digital Module Scheme because it is more practical to his condition. On the other hand, his mobile phone notified him that Module 5 of Statistics and Probability had been posted to their Facebook Learning Space, ready to be accessed and downloaded. Surprisingly, there's a problem with Harvey. He forgot his 5-digit smartphone passcode! Let us help Harvey decode his passcode by reviewing the past lesson on the introduction of sampling distribution of sample means. Direction: On your answer sheets, copy the Code Table and review the concepts of the previous lesson by completing the paragraphs and tables provided. Link them to the corresponding number in the Code Table. Code Table: CODE ANSWER

602.720

3.727

Concept 1 Reviewer:

24.550

13.889

5.860

Recall that the Variance (𝛔𝟐) and the Standard deviation (𝛔) of ungrouped data are computed by using the formulas presented on the next page, respectively.

4

𝝈𝟐 =

𝚺𝑿𝟐 𝑵

𝚺𝐗 𝟐

−( ) 𝑵

𝝈=√

where:

𝚺𝑿𝟐 𝑵

𝚺𝐗 𝟐

−( ) 𝑵

where:

σ2 = variance

σ = standard deviation

X = score or value

X = score or value

N = number of scores or values

N = number of scores or values

Concept 1 Example: Given the set of data: X = { 2, 5, 6, 9, 11, 13 }, complete the corresponding table and compute for the variance and standard deviation. X

X2

2

4

5

25

6

36

9

81

11

121

13

169

46

436

Variance: 𝜎2 =

𝜎2 =

Σ𝑋 2 𝑁

436 6

Standard Deviation: ΣX 2

𝜎=√

−( ) 𝑁

46 2

𝜎=√

−( ) 6

𝜎 2 = (𝟏)

Σ𝑋 2 𝑁

436 6

ΣX 2

−( ) 𝑁

46 2

−( ) 6

𝜎 = (𝟐)

Concept 2 Reviewer: Recall that the Mean (𝛍) or the Expected Value E(X) of a discreet probability distribution is computed using the formula presented below. 𝝁 = 𝑬(𝑿) = 𝚺[𝑿 ∙ 𝑷(𝑿)]

5

where:

μ = mean

E(X) = expected value X = value of the random variable P(X) = probability value of the random variable

Concept 2 Example: Find the mean or the expected value of the given probability distribution below. X

P(X)

X • P(X)

0

0.100

0.000

2

0.150

0.300

4

0.200

0.800

5

0.140

0.700

6

0.150

0.900

8

0.090

0.720

11

0.030

0.330

15

0.050

0.750

14

0.050

0.700

16 18

0.030 0.010

0.480 0.180

46

1.000

5.860

Mean or Expected Value:

𝜇 = 𝐸(𝑋) = Σ[𝑋 ∙ 𝑃 (𝑋)]

𝜇 = 𝐸(𝑋) = (𝟑)

Concept 3 Reviewer:

Recall that the Variance (𝛔𝟐) and the Standard deviation (𝛔) of probability distribution are computed by using the formulas presented below, respectively. 𝝈𝟐 = 𝚺 [𝑿𝟐 ∙ 𝑷(𝑿) ] − 𝝁𝟐

𝝈 = √𝚺 [𝑿𝟐 ∙ 𝑷(𝑿)] − 𝝁𝟐

6

where:

where:

σ2 = variance

σ = standard deviation

X = score or value

X = score or value

P(X) = probability value of the random variable μ = mean or expected value

P(X) = probability value of the random variable μ = mean or expected value

Concept 3 Example: Compute the variance and standard deviation of the given probability distribution below. X

X2

P(X)

X2 • P(X)

0 2

0 4

0.100 0.150

0.000 1.200

4

16

0.200

12.800

5 6

25 36

0.140 0.150

17.500 32.400

8

64

0.090

46.080

11

121

0.030

39.930

15

225

0.050

168.750

14

196

0.050

137.200

16

256

0.030

122.880

18

324

0.010

58.320

46

1 267

1.000

637.060

Variance:

Standard Deviation:

𝜎 2 = Σ [𝑋 2 ∙ 𝑃(𝑋)] − 𝜇2

𝜎 = √Σ [𝑋 2 ∙ 𝑃(𝑋)] − 𝜇2

𝜎 2 = 637.060 − (5.860)2

𝜎 = √637.060 − ( 5.860)2

𝜎 2 = (𝟒)

𝜎 = (𝟓)

Congratulations! You accessed Harvey’s smartphone. Now, Module 5 of his subject, Statistics and Probability, can be made available and downloaded. Let us continue to help Harvey explore his learning experience through this module!

7

The next part of this module will be a brief and simple discussion of the lesson. This will be followed by a series of formative assessment activities. Hey, just boost it up!

What is It It’s time to take a general tour around this module. This portion of your journey acts a simple and brief discussion of the lesson that aims to help you discover and understand new concepts and skills. This will really help you understand the real-life applications of mean and variance of sampling distribution of the sample means. Basically, the flow of the discussion will be starting with the presentation of the example problem about the mean of the sampling distribution of the sample means and to be followed by the corresponding problem about its variance and standard deviation. By helping Harvey in this module, you are able to have a learning buddy in order for you to reach your goals in finishing this module with learned skills and competencies on your intellects! First, Harvey wants to investigate the mean of the sampling distribution of the sample means and compare it with the mean of the population. After the comparison of the means, Harvey must compare the variance and the standard deviation of the population to the sample means. Let us do this!

Illustrative Example: The following table gives the sum of tutorial rate of six teachers in Central Luzon per month. Suppose that random samples of size 4 are taken from this population of six teachers, do the following tasks. Teacher

Tutorial Rate (in thousand pesos) X

A

8

B

12

C

16

D

20

E

24

F

28

8

Solve for the mean of the population μ. Solve for the mean of the sampling distribution of the sample means μx. Compare μ and μx . Solve for the variance (σ2 ) and the standard deviation (σ) of the population. 5. Solve the variance (σ2 x ) and the standard deviation (σx ) of the sampling distribution of the sample means μx . 6. Compare σ and σx . 1. 2. 3. 4.

Solutions: 1. The population mean μ is solved as follows. ΣX 8 + 12 + 16 + 20 + 24 + 28 108 = = 𝜇= = 18 6 𝑁 6 Therefore, the population mean of the tutorial rates of the select teachers in Central Luzon is 18 thousand pesos per month. 2. To solve for the mean of the sampling distribution of the sample means, the following steps are to be considered. a. Identify the possible samples of size 4 and compute their individual means.

Possible Sample 8, 12, 16, 20

Sample Mean  𝐗 14

8, 12, 16, 24

15

8, 12, 16, 28

16

8, 12, 20, 24

16

8, 12, 20, 28

17

8, 12, 24, 28 8, 16, 20, 24

18 17

8, 16, 20, 28

18

8, 16, 24, 28

19

8, 20, 24, 28

20

12, 16, 20, 24

18

12, 16, 20, 28

19

12, 16, 24, 28

20

12, 20, 24, 28

21

16, 20, 24, 28

22

9

b. Construct the sampling distribution table for the sample means and multiply the sample means to their probabilities. Sample Mean  𝐗

Frequency F

14

1

15

Probability ) P (𝐗

• P (𝐗 ) 𝐗

1/15

14/15

1

1/15

15/15

16

2

2/15

32/15

17

2

2/15

34/15

18

3

3/15

54/15

19

2

2/15

38/15

20

2

2/15

40/15

21

1

1/15

21/15

22

1

1/15

22/15

Total

15

15/15 or 1

270/15 = 18
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