Statistics Probability Q3 Mod3 The Normal Distribution PDF

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Statistics andProbabilityQuarter 3 – Module 3:The Normal DistributionStatistics and Probability Alternative Delivery Mode Quarter 3 – Module 3: The Normal Distribution First Edition, 2020Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Phili...

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Statistics and Probability Quarter 3 – Module 3: The Normal Distribution

Statistics and Probability Alternative Delivery Mode Quarter 3 – Module 3: The Normal Distribution 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 3: The Normal Distribution

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.

What I Need to Know

This module was designed and written to help you understand the concept of the normal distribution. The scope of this module helps you to appreciate and understand learning situations that can be used in a day-to -day basis experience. The language used in this learning kit recognized the diverse vocabulary level of students for a higher understanding of the lesson. The lessons were arranged to follow the standard sequence of the course. However, the manner in which you read them can be possibly changed to correspond with the textbook that you might be using now. Thus, this module is divided into two lessons, respectively as follows • •

Lesson 1 – The Normal Distribution and Its Properties Lesson 2 – The Standard Normal Distribution

After going through this module, you are expected to: 1. illustrate a normal random variable and its characteristics (M11/12SP-IIIc1); 2. identify regions under the normal curve that correspond to different standard normal values (M11/12SP-IIc-3); 3. convert a normal random variable to a standard normal variable and vice versa (M11/12SP-IIIc-4); and 4. compute probabilities and percentiles using the standard normal distribution (M11/12SP-IIIc-d-1).

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What I Know

Directions: Choose the best answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following illustrations represents normal distribution? A.

C.

B.

D.

2. What is another name for normal distribution? A. Gaussian distribution B. Poisson distribution C. Bernoulli’s distribution D. Probability distribution 3. What is the total area in the distribution under the normal curve? A. 0 B. 1 C. 2 D. 3 4. Which of the following is a parameter of normal distribution? A. mean B. standard deviation C. mean and standard deviation D. none of the above 5. The graph of a normal distribution is symmetrical about the ________. A. mean B. standard deviation C. mean and standard deviation D. none of the above 6. What percent of the area under a normal curve is within 2 standard deviations? A. 68.3% B. 95.4% C. 99.7% D. 100%

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7. How many standard deviations are there in each inflection point? A. 0 B. 1 C. 2 D. 3 8. Which of the following denotes the standard normal distribution? A. A B. X C. Y D. Z 9. A random variable X has a mean of 12 and a standard deviation of 3.2. What is the corresponding z-score for x = 8? A. -1.25 B. -1.50 C. -1.75 D. -2.25 10. What is the area under the normal curve if the z-score given is 2.14? A. 0.9830 B. 0.9834 C. 0.9838 D. 0.9842 11. What is the z-score if the area under the normal curve is 0.0475? A. -1.66 B. -1.67 C. 1.66 D. 1.67 12. What is the area under the normal curve between the z-scores -1.99 and 1.56? A. 0.9173 B. 0.9317 C. 0.9369 D. 0.9639 13. What is the area of the P(Z > 2.58) using the standard normal curve? A. 0.0049 B. 0.0051 C. 0.9949 D. 0.9951 14. In an achievement test, the mean score of normally distributed values is 70 and the standard deviation is 12. What is the percentage of students who got a score of 85 and above? A. 0.1056 B. 0.1075 C. 0.8925 D. 0.8944 15. What is the value of 70th percentile in a standard normal distribution? A. 0.51 B. 0.52 C. 0.61 D. 0.62

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Lesson

1

The Normal Distribution and Its Properties

The normal distribution is just one of the distributions to be discussed in this course. It is also considered as the most important distribution in Statistics because it fits many real-life situations. This lesson will bring us a deeper understanding of the normal distribution and its characteristics.

What’s In Directions: Anticipation-Reaction Guide. Complete the table by putting check as to AGREE or DISGAREE on the corresponding columns for the following statements. Write your answers on a separate sheet of paper. Statement

Agree

1. The normal curve of the distribution is bell-shaped. 2. In a normal distribution, the mean, median and mode are of equal values. 3. The normal curve gradually gets closer and closer to 0 on one side. 4. The normal curve is symmetrical about the mean. 5. The distance between the two inflection points of the normal curve is equal to the value of the mean.

Notes to the Teacher This contains helpful tips and strategies that will help you in guiding the learners.

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Disagree

What’s New Before proceeding to our discussion, let us consider this activity that will give us ideas about our lesson. Consider the random event of tossing four coins once, then follow these steps: 1. 2. 3. 4. 5.

List all the possible outcomes using the tree diagram. Determine the sample space. Determine the possible values of the random variables. Assign probability values P(X) to each of the random variable. Construct a probability histogram to describe the P(X). Answer the following guide questions:

1. How many possible outcomes are there? 2. What composes the sample space? 3. How will you describe the histogram?

What is It To give us a deeper understanding of the concept of the normal distribution, let us learn more about its properties. The following are the properties that can be observed from the graph of a normal distribution, also known as Gaussian distribution. 1. The graph is a continuous curve and has a domain -∞ < X < ∞. • This means that X may increase or decrease without bound. 2. The graph is asymptotic to the x-axis. The value of the variable gets closer and closer but will never be equal to 0. • As the x gets larger and larger in the positive direction, the tail of the curve approaches but will never touch the horizontal axis. The same thing when the x gets larger and larger in the negative direction. 3. The highest point on the curve occurs at x = µ (mean). • The mean (µ) indicates the highest peak of the curve and is found at the center. • Take note that the mean is denoted by this symbol µ and the standard deviation is denoted by this symbol .

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The median and mode of the distribution are also found at the center of the graph. This indicates that in a normal distribution, the mean, median and mode are equal. The curve is symmetrical about the mean. • This means that the curve will have balanced proportions when cut in halves and the area under the curve to the right of mean (50%) is equal to the area under the curve to the left of the mean (50%). The total area in the normal distribution under the curve is equal to 1. • Since the mean divides the curve into halves, 50% of the area is to the right and 50% to its left having a total of 100% or 1. In general, the graph of a normal distribution is a bell-shaped curve with two inflection points, one on the left and another on the right. Inflection points are the points that mark the change in the curve’s concavity. • Inflection point is the point at which a change in the direction of curve at mean minus standard deviation and mean plus standard deviation. • Note that each inflection point of the normal curve is one standard deviation away from the mean. Every normal curve corresponds to the “empirical rule” (also called the 68 95 - 99.7% rule): •

4.

5.

6.

7.

•

about 68.3% of the area under the curve falls within 1 standard deviation of the mean

•

about 95.4% of the area under the curve falls within 2 standard deviations of the mean

•

about 99.7% of the area under the curve falls within 3 standard deviations of the mean.

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Consider the following examples: 1. Suppose the mean is 60 and the standard deviation is 5, sketch a normal curve for the distribution. This is how it would look like.

2. A continuous random variable X is normally distributed with a mean of 45 and standard deviation of 6. Illustrate a normal curve and find the probability of the following: a. P (39 < X < 51) = 68.3%

c. P (X > 45) = 50%

*Since the area covered is 1 standard of the deviation to the left and to the right. b. P (33 < X < 63) = 97.55%

* Since the area covered is half curve. d. P (X < 39) = 15.85%

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What’s More Directions: Read the following statements carefully. Write ND if the statement describes a characteristic of a normal distribution, and NND if it does not describe a characteristic of a normal distribution. Write your answers on a separate sheet of paper. 1. 2. 3. 4. 5.

The curve of the distribution is bell-shaped. In a normal distribution, the mean, median and mode are of equal values. The normal curve gradually gets closer and closer to 0 on one side. The curve is symmetrical about the mean. The distance between the two inflection points of the normal curve is equal to the value of the mean. 6. A normal distribution has a mean that is also equal to the standard deviation. 7. The two parameters of the normal distribution are the mean and the standard deviation. 8. The normal curve can be described as asymptotic. 9. Two standard deviations away from the left and right of the mean is equal to 68.3%. 10. The area under the curve bounded by the x-axis is equal to 1.

What I Have Learned Now that you have learned the concept of normal distribution, you may proceed to the next activity. Direction: Complete the given diagram below by filling up the necessary details about normal distribution. List 5 properties of normal distribution

NORMAL DISTRIBUTION

State the empirical rule.

Construct a normal curve.

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What I Can Do Directions: Make a sketch for each of the 3 areas under the normal curve as stated in the empirical rule. Using a mosaic art, shade the area that corresponds to the area under the normal curve. You may use eggshells, old magazines, dried leaves or any materials available at home.

Assessment Directions: Read the instructions given and write your answers on your answer sheets. A. Multiple Choice. Choose the letter of the best answer. 1. What is another name for normal distribution? A. Gaussian distribution B. Poisson distribution C. Bernoulli’s distribution D. Probability distribution 2. What is the total area in the distribution under the curve? A. 0 B. 1 C. 2 D. 3 3. What marks the change in the curve’s concavity? A. curve B. inflection points C. mean D. standard deviation 4. Which value is found at the center of the normal curve? A. mean B. median C. mode D. all of the above 5. Which of the following is a parameter of normal distribution? A. mean B. standard deviation C. mean and standard deviation D. none of the above

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6. Which of the following symbols is used to denote the mean? A. B. µ C. α D. ∞ 7. Which of the following does not describe a normal curve? A. asymptotic B. bell-shaped C. discrete D. symmetrical about the mean 8. What percent of the area under a normal curve is within 2 standard deviations? A. 68.3% B. 95.4% C. 99.7% D. 100% 9. What percent of the area under a normal curve is within 1 standard deviation? A. 68.3% B. 95.4% C. 99.7% D. 100% 10. What percent of the area under a normal curve is within 3 standard deviations? A. 68.3% B. 95.4% C. 99.7% D. 100% B. Read, analyze, and answer the given involving normal curve. 1. A continuous random variable X is normally distributed with a mean of 56.3 and standard deviation of 7.2. Illustrate a normal curve and find its probability.

P (34.7 < X < 63.5)

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Additional Activities Distribution of Balls in a Quincunx In this link, http://www.mathsisfun.com/data/quincunx.html, the webpage will show a quincunx or "Galton Board" (named after Sir Francis Galton). This has a triangular array of pegs wherein balls are dropped onto the top peg. Subsequently, balls bounce their way down to the bottom where they are collected in little bins. Each time a ball hits one of the pegs, it bounces either to the left or right with equal probability. As a result, the number of pegs collecting in the bins form a bell-shaped curve, especially as the number of rows (and bins) as well as the number of balls increases.

0 1 2 3 4 5 6 (adopted from the Teaching Guide for Senior High School – Statistics and Probability, pages 172-173)

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Lesson

2

The Standard Normal Distribution

The standard normal distribution, which is denoted by Z, is also a normal distribution having a mean of 0 and a standard deviation of 1. Since the normal distribution can have different values for its mean and standard deviation, it can be standardized by setting the µ = 0 and the = 1.

What’s In Direction: On your answer sheets, write the area corresponding to the shaded part of the normal curve.

1.

2.

3.

Notes to the Teacher This contains helpful tips and strategies that will help you in guiding the learners.

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What’s New

Directions: Observe the illustrations then answer the questions that follow. Write your answers on a separate sheet of paper.

Figure A

Figure B

Guide Questions: 1. What is the mean? Figure A: _______________ Figure B: _______________ 2. What is the standard deviation? Figure A: _______________ Figure B: _______________ 3. What is the area of the shaded region? Figure A: _______________ Figure B: _______________ 4. What did you do to identify the area of the shaded region? Figure A: _______________________

Figure B: ______________________

5. Did you use the same method? ________________________

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What is It As mentioned earlier, normal variable is standardized by setting the mean to 0 and standard deviation to 1. This is for the purpose of simplifying the process in approximating areas for normal curves. As shown below is the formula used to manually compute the approximate area.

f(z) =

1 √2𝜋

𝑧2

𝑒− 2

However, this formula is seldom used because a table was created to summarize the approximate areas under the standard normal curve and to further simplify the process. This table of probabilities is known as the z- table. The Z - Table Let us get a closer look at the z-table. The outermost column and row represent the z-values. The first two digits of the z-value are found in the leftmost column and the last digit (hundredth place) is found on the first row. Suppose the z-score is equal to 1.85, locate the first two digits 1.8 in the leftmost column and the last digit, .05, can be located at the first row. Then find their intersection which gives the corresponding area. Therefore, given z = 1.85, the area is equal to 0.9678. z 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.00 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772

0.01 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778

0.02 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783

0.03 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788

0.04 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793

0.05 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798

0.06 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803

0.07 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808

0.08 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812

0.09 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817

Other examples are as follow: 1. Find the area that corresponds to z = 2.67 z 2.4 2.5 2.6 2.7 2.8 2. Find

0.00 0.9918 0.9938 0.9953 0.9965 0.9974 the area

0.01 0.02 0.03 0.9920 0.9922 0.9925 0.9940 0.9941 0.9943 0.9955 0.9956 0.9957 0.9966 0.9967 0.9968 0.9975 0.9976 0.9977 that corresponds to z

0.04 0.9927 0.9945 0.9959 0.9969 0.9977 = 1.29

3. Find the area that corresponds to z = 3

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Answer: 0.9962 0.05 0.9929 0.9946 0.9960 0.9970 0.9978