Hybrid Statistic and probability 11 Stat Q3 W4 PDF

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QUARTER 3: WEEK 6 Department of Education Republic of the Philippines Introductory Message For the facilitator: This module was collaboratively designed, developed and evaluated the Development and Quality Assurance Teams of SDO TAPAT to assist you in helping the learners meet the standards set the ...

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QUARTER 3: WEEK 6

Introductory Message For the facilitator: This module was collaboratively designed, developed and evaluated by the Development and Quality Assurance Teams of SDO TAPAT to assist you in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. As a facilitator, you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.

For the learner: This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. The following are some reminders in using this module: 1. Use the module with care. Use a separate sheet of paper in answering the exercises. 2. Don’t forget to answer Let’s Try before moving on to the other activities included in the module. 3. Read the instruction carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!

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LET’S LEARN This module was designed and written to help you master sampling distributions of sample means. 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 – Sampling Distributions of Sample Means. Lesson 2 – Find the Mean, Variance, and Standard Deviation of the Sample Means.

After going through this module, you are expected to: 1. identify sampling distributions of statistics (sample mean), and 2. find the mean and variance of the sampling distribution of the sample mean.

LET’S TRY Directions: Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. A probability distribution that describes the probability for each mean of all the samples with the same sample size n is called____________. A. Frequency Distribution C. Sampling Distribution B. Normal Distribution D. Non-probability 2. A population consists of values 12, 14, 16, 18. What is the population mean? A. 14.0 C. 15 B. 16.5 D. 14.5 3. A sample consists of values 8, 16, 24. What is the sample mean? A.16 C. 19 B. 17 D. 13.33 4. If the population standard deviation 𝜎 of a dataset is 7, what is the population variance? A. 30 C. 40 B. 25 D. 49 5. If the population variance of a set of data is 25, what is the population standard deviation? A. 3.5 C. 4.5 B. 5.0 D. 6.0

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6. If a population has a mean of 4.6, what is the mean of the sampling distribution of its means? A. 4.6 C. 6.6 B. 4.6 D. 7.6 7. If a population has a mean of 15, what is the mean of the sampling distributions of its means? A. 20 C. 30 B. 15 D. 35 8. If the mean of the sampling distribution of the means is 12.4, what is the mean of the population? A. 12.0 C. 12.4 B. 12.2 D. 12.6 9. It states that, “If samples of size 𝑛, where n is sufficiently large, are drawn from any population with a mean µ and standard deviation 𝜎 , then the sampling distribution of sample means approximates a normal distribution.” A. Probability Distribution C. Central Limit Theorem B. Normal Distribution D. Random Variable. 10. If a population has a variance of 9.3, what is the variance of the sampling distribution of its means if 𝑛 = 3. A. 3.1 C. 3.3 B. 3.2 D. 3.4

Alternatively, you may answer these questions online! Use this link on your cellphone, laptop or desktop:

Write your score here

https://tinyurl.com/STATQ3W6TRY Use proper capitalization to activate the link. Make sure you are connected to the internet! You will see your score after completing the test. Make sure to screenshot your work as proof to your teacher then write your score in the box.

Lesson

1

Sampling Distributions

When you select a random sample from a population, the numerical descriptive measures you calculate from the sample are called statistics. These statistics vary or change

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from each different random sample you select; that is, they are random variables. The probability distributions for statistics are called sampling distributions.

LET’S RECALL Activity 1.1 Suppose that a population consists of the six (6) employees in a small business. The random variable of interest 𝑥, is the number of years the employee has been with the business. The values of the variable are as follows:

Employees

Years in the Business 1

Shiela

2

Claudine

3

Mildred

4

Divine

5

Bridgette

6

Sophia

What is the population mean, population variance and population standard deviation of the given data? Solution Given the vales of x = 1,2,3,4,5,6. We first solve for the population mean, we get µ=

∑𝑥 𝑁

=

1+2+3+4+5+6 21 = = 6 6

3.5

Then we solve for the population variance and standard deviation as shown on the table below. Employees

Year in the Business (𝑥)

𝑥−µ

(𝑥 − µ)²

Shiela

1

-2.5

6.25

Claudine

2

-1.5

2.25

Mildred

3

-0.5

0.25

Divine

4

0.5

0.25

Bridgette

5

1.5

2.25

Sophia

6

2.5

6.25

Total

21

17.50

5

𝜎² =

∑(𝑥−µ)²

𝜎=√

𝑁

∑(𝑥−µ)² 𝑁

=

17.50 6

=√

= 2.92

17.50 6

= √2.92 = 1.71

Therefore, the population mean is 3.5, the population variance is 2.92, and the population standard deviation is 1.71.

LET’S EXPLORE Activity 1.2 Suppose that a population consists of the six (6) employees in a small business. The random variable of interest 𝑥, is the number of years the employee has been with the business. The values of the variable are as follows:

Employees

Years in the Business 1

Shiela

2

Claudine

3

Mildred

4

Divine

5

Bridgette

6

Sophia

What is the sampling distribution of the sample means for a sample of size 2?

LET’S ELABORATE Steps in Constructing a Sample Distribution 1. Find the number of possible samples that can be drawn from the finite population using combination formula. 𝑁! 𝐶𝑛 = (𝑁−𝑛 )!𝑛!

𝑁

2. Randomly draw all possible sample of size n from a discrete, finite population of size N. 3. List observed values of the statistic and its corresponding frequency of occurrence.

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Solution to Activity 1.2 To generate the sampling distribution of the sample means, we need to collect all possible sample size without replacement from the population. Step 1. By applying the combination formula, we can determine the number of possible samples of size 2. Note that N=6 and n=2. 6 C2

=

6!

(6−2)!2!

=

720 = 48

15

There are 15 possible samples of size 2. Step 2. Now we list the 15 distinct samples of size 2 which can be drawn from the population. Table 1: All Possible Samples of Size 2

Observations

Employees

Years

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Shiela, Claudine Shiela, Mildred Shiela, Divine Shiela, Bridgette Shiela,Sophia Claudine, Milded Claudine, Divine Claudine, Bridgette Claudine, Sophia Mildred, Divine Mildred, Bridgette Mildred, Sophia Divine, Bridgette Divine, Sophia Bridgette, Sophia

1,2 1,3 1,4 1,5 1,6 2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6

The table shows the sampling distribution of the population consists of six (6) employees in a small business of sample means for the sample of size 2.

For more learning exploration, visit and watch the following

channels!

Sampling Distribution of the Sample Mean by JBStatistics

https://www.youtube.com/watch?v=0zqNGDVNKgA

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LET’S DIG IN Activity 1.3 There are four newly hired salespersons at Honda Cars Taguig Branch. The number of motorcycle sold by each newly hired salesperson last month is 4.5, 8, and 9. a. Complete the information needed in the table below. b. How many samples of size 2 are possible? c. List all possible samples of size 2. Observation

Sample

LET’S REMEMBER Steps in Constructing a Sample Distribution 1. Find the number of possible samples that can be drawn from the finite population using combination formula. 𝑁𝐶𝑛 =

𝑁! (𝑁−𝑛)!𝑛!

2. Randomly draw all possible sample of size n from a discrete, finite population of size 𝑁. 3. List observed values of the statistic and its corresponding frequency of occurrence.

LET’S APPLY Activity 1.4 Suppose that each of the five workers making up a population of support personnel of a fabrication department. The number of defective products by each worker is presented in the table.

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Workers

Rowel

Rudolph

Edwin

Jerry

Vincent

No. of Defective Products

3

1

2

4

6

a. How many samples of size 2 are possible? b. List all possible samples of size 2.

Lesson

2

Mean, Variance, and Standard Deviation of Sampling Distributions of the Sample Mean

There are many different possible samples of the same size that can be drawn from a given population. A statistic such as mean, variance and standard deviation can be computed for each of the samples drawn.

LET’S RECALL Activity 2.1 Consider the population consisting of the values 2,3, and 4. List all the possible samples of size 2 that can be drawn from the population with replacement. Then compute the mean 𝑥 for each sample. Solution All possible samples of size 2 with replacement are listed in the second column. The corresponding means of all these samples are shown in the third column. For instance, the first sample consists of (2, 2) and the corresponding mean is (2 + 2) ÷ 2 = 2.0. Means of Samples Drawn with Replacement from the Population = (2,3,4) Observation 1 2 3 4 5 6 7 8 9

Sample (2,2) (2,3) (2,4) (3,2) (3,3) (3,4) (4,2) (4,3) (4,4)

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2.0 2.5 3.0 2.5 3.0 3.5 3.0 3.5 4.0

𝑥

A total of 9 samples with 9 sample means can be drawn from the population (2,3, and 4).

LET’S EXPLORE Activity 2.2 Consider the population consisting of values 2, 4, and 6 . a. List all the possible samples of size 2 that can be drawn from the population with replacement. b. Find the mean of the sampling distribution of means. c. Find the variance of the sampling distribution of the means. d. Find the standard deviation of the means.

LET’S ELABORATE To find the mean µ𝑥 of the sampling distribution of means is:

µ𝑥 =

where

∑ 𝑥 𝑛

𝑥 is the sample mean

𝑛 is the total number of observations

To find the variance 𝜎²𝑥 of the sampling distribution of means is:

𝜎²𝑥 = where

∑( 𝑥 − µ𝑥 )² 𝑛

𝑥 is the sample mean

µ𝑥 is the mean of the sampling distribution of the means. 𝑛 is the total number of observations

To find the standard deviation 𝜎𝑥 of the sampling distribution of means is: 𝜎𝑥 = √

∑( 𝑥 − µ𝑥 )² 𝑛

Where 𝜎𝑥 is the standard deviation of the sampling distribution of the means. 𝑥 is the sample mean

µ𝑥 is the mean of the sampling distribution of the means.

𝑛 is the total number of observations

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Solution to Activity 2.2 a. Means of Samples Drawn with Replacement from the Population = (2, 4, 6) are listed in the second column of the table below. Observation

Sample

𝑥

𝑥 - µ𝑥

(𝑥 - µ𝑥 )²

1

(2,2)

2.0

-2.0

4.0

2

(2,4)

3.0

-1.0

1.0

3

(2,6)

4.0

0

0

4

(4,2)

3.0

1.0

1.0

5

(4,4)

4.0

0

0

6

(4,6)

5.0

1.0

1.0

7

(6,2)

4.0

0

0

8

(6,4)

5.0

1.0

1.0

9

(6,6)

6.0

2.0

4.0

∑𝑥 = 36

∑(𝑥 - µ𝑥 )²= 12

b. To solve the mean of the sampling distribution of the means use the formula:

µ𝑥 = µ𝑥 =

∑ 𝑥 𝑛

36 9

=

4.0

Thus, the mean of the sampling distribution of the means is 4.0. c. To solve the variance of the sampling distribution of the means use the formula:

𝜎²𝑥 = 𝜎²𝑥 =

∑( 𝑥 − µ𝑥 )² 𝑛

12 9

=

1.33

Thus, the variance of the sampling distribution of the means is 1.33.

d. To solve the standard deviation of the sampling distribution of the means use the formula:

𝜎𝑥 = √

∑( 𝑥 − µ𝑥 )² 𝑛

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𝜎𝑥 = √ 9 = 1.15 Thus, the standard deviation of the sampling distribution of the means is 1.15

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For more learning exploration, visit and watch the following

channels!

Mean and Standard Deviation of the sampling distribution in Filipino by Numberbender

https://www.youtube.com/watch?v=0Rxx1hb5Mmc

LET’S DIG IN Activity 2.3 Complete the following table. Number 1 2 3 4 5

Mean 𝑥

Sample (2,2) (2,4) (2,6) (2,8) (4,4)

𝑥 - µ𝑥

(𝑥 - µ𝑥 )²

Calculate the following: a. Mean of the sampling distribution of means. b. Variance of the sampling distribution of means. c. Standard Deviation of the sampling distribution of means.

LET’S REMEMBER ➢ The formula to find the mean µ𝑥 of the sampling distribution of means is:

µ𝑥 =

∑ 𝑥 𝑛

➢ The formula to find the variance 𝜎²𝑥 of the sampling distribution of means is:

𝜎²𝑥 =

∑( 𝑥 − µ𝑥 )² 𝑛

➢ The formula to find the standard deviation 𝜎𝑥 of the sampling distribution of means is:

𝜎𝑥 = √

∑( 𝑥 − µ𝑥 )² 𝑛

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LET’S APPLY Activity 2.4 A population consists of four numbers (2,4,6,8). Consider all possible samples of size 2 that can be drawn with replacement from this population. Find the following: a. Mean of the sampling distribution of means. b. Variance of the sampling distribution of means. c. Standard Deviation of the sampling distribution of means

LET’S EVALUATE (Post-test) Directions: Choose the letter of the best answer. 1. Which of the following statements is correct? A. The mean of the sampling distribution of the means is greater than the population mean. B. The mean of the sampling distribution of the sample means is less than the population mean. C. The means of the samples drawn from a population are always equal to the population mean. D. The means of the samples drawn from a population may be equal, less than or greater than the population mean. 2. A population consists of values 3,6,9, and 12. What is the population mean? A. 5.5 C. 7.5 B. 6.5 D. 8.5 3. A sample consists of values 4, 8, 12, and 16. What is the sample mean? A.10 C. 12 B. 11 D. 13 4. If the population standard deviation of a dataset is 4, what is the population variance? A. 9 C. 16 B. 12 D. 25 5. If the population variance of a set of data is 25, what is the population standard deviation? A. 5.0 C. 6.0 B. 5.5 D. 6.5

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6. If a population has a mean of 6.5, what is the mean of the sampling distribution of its means? A. 6.0 C. 7.0 B. 6.5 D. 7.5 7. If a population has a mean of 15, what is the mean of the sampling distributions of its means? A. 10 C. 20 B. 15 D. 25 8. If the mean of the sampling distribution of the means is 13.5, what is the mean of the population? A. 12.5 C. 13.5 B. 13.0 D. 14.0 9.If samples of size n, where n is sufficiently large, are drawn from any population with a mean µ and standard deviation, then the sampling distribution of sample means approximates a normal distribution. A. Probability Distribution C. Central Limit Theorem B. Normal Distribution D. Random Variable 10. If a population has a variance of 9.3, what is the variance of the sampling distribution of its means? The sampling distribution was derived with the sample size 𝑛 = 3 and all possible samples area drawn with replacement. A. 2.9 C. 3.1 B. 3.0 D. 3.2

Alternatively, you may answer these questions online ! Use this link on your cellphone, laptop or desktop:

Write your score here

https://tinyurl.com/STATQ3W6EVALUATE Use proper capitalization to activate the link. Make sure you are connected to the internet! You will see your score after completing the test. Make sure to screenshot your work as proof to your teacher then write your score in the box.

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References Statistics and Probability: Next Century Mathematics, Jesus P. Mercado and Fernando B. Orines. Statistics and Probability: MSA, Merle S. Alferez and Ma. Cecilia A. Duro. Ched-Deped SHS Statistics and Probability TG Acebedo. Introducing to Probability and Statistics, 10th Edition, William Mendenhall, Robert J. Beaver and Barbara M. Beaver. Worktext in Statistics, Florite O. Vizcarra and Eduardo G. Lubina

Development Team of the Module Writer: VINCENT M. TAGUINOD Editors: CONTENT EVALUATORS: LAMBERT QUESADA LANGUAGE EVALUATOR: AILEEN GENOSO

Reviewers: MRS. MIRASOL I. RONGAVILLA ARMANDO V. EROLIN Illustrators: Layout Artist: Management Team:

DR. MARGARITO B. MATERUM, SDS DR. GEORGE P. TIZON, SGOD-Chief DR. ELLERY G. QUINTIA, CID Chief MRS. MIRASOL I. RONGAVILLA, EPS - MATH DR. DAISY L. MATAAC, EPS – LRMS/ ALS

For inquiries, please write or call: Schools Division of Taguig city and Pateros Upper Bicutan Taguig City Telefax: 8384251 Email Address: [email protected]

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Activity 3.3

Activity 1.4

Assessment

a. 4.0 b. 1.33 c. 1.15

a. 10

1.C 2.C 3.A 4.D 5B 6.B 7.B 8.C 9.C 10.A

...