Solving Problems Involving Normal Distributions, Cheat Sheet of Mathematics

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Published by the Learning Resource Management and Development System (LRMDS) Department of Education Region VI – Western Visayas DIVISION OF CAPIZ Roxas City Copyright 2017 “Section 9 of the Presidential Decree No. 49 provides”

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ANALYN N. CALITINA MARY JOY V. DIAZ ROGER B. CORROS Developer/Writer Illustrator Graphic Artist Quality Assured by: RAMONA C. IBANEZ, ROWENA F. LUZA SSHT VI MT-II FELSIE D. OBUYES HT – III ELENIA P. BARANDA SHIRLEY A. DE JUAN EPS-Mathematics EPS Learning Resource Management Approved for the use of the Schools Division:

SEGUNDINA F. DOLLETE, Ed.D. Chief-Curriculum Implementation Division NICASIO S. FRIO Acting Schools Division Superintendent MIGUEL MAC D. APOSIN, Ed.D.; CESO V Schools Division Superintendent Grade Level: II – Probability and Statistics Language: English This first digital edition has been produced for print and online distribution within the Department of Education, Philippines via the Learning Resources Management Systems (LRMDS) Portal Region VI.

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TOPIC: SOLVING PROBLEMS INVOLVING NORMAL DISTRIBUTIONS

The normal distribution is a type of data distribution that is observed in a lot of instances in real life. There are many continuous random variables that we measure in everyday instances that have a defined normal range such as IQ scores, blood pressure, and test scores.

A normal distribution can have different variations depending on its mean and standard deviation. It will be convenient to analyze only a normal distribution with a mean of 0 and a standard deviation of 1. This normal distribution is called standard normal distribution. By converting the possible values of a normally distributed random variable to its corresponding z-score, the normal distribution is standardized, thus making it easier to analyze.

The z-score is not entirely a different quantity. It is derived from X, which is the value that the continuous random variable assumes. To determine how the z-score is related to any value of a random variables use this formula:

Competencies: (M11/12SP-IIIc 1-3)  Illustrates a normal random variable and its characteristics.  Construct a normal curve.  Identifies regions under the normal curve corresponding to different standard normal values.

To solve for the probability in a normal distribution, we cannot just shade the area bounded by the curve. To make this a lot easier, the normal distribution is converted into a standard normal distribution.

Illustrative Examples:

1. The number of tricycles that pass through Roxas Avenue everyday resembles a normal distribution with a mean of 7410 and a standard deviation of 688. On a given day, what is the probability that the number of tricycles that will pass through Roxas Avenue will be between 6000 and 7000 tricycles?

Example 2. Refer to problem Number 1, what is the entrance score result of a student who has a z-score of -2?

Given: 𝜇 = 75 𝜎 = 2. 5 𝑧 = − 2 Solution: 𝑧 = 𝑋− 𝜎^ 𝜇 Equate the formula to solve for x 𝑋 = 𝑧𝜎 + 𝜇 𝑋 = (− 2 )( 2. 5 ) + 75 𝑋 = − 5 + 75 𝑿 = 𝟕𝟎

Solution: Given: 𝜇 = 7410 𝜎 = 688 𝑥 1 = 6000 𝑥 2 = 7000

𝑧 1 = 𝑋 − 𝜇𝜎 =^6000688 −^^7410 = − 2. 05

𝑧 2 = 𝑋 − 𝜇𝜎 =^7000688 −^^7410 = − 0. 60

The probability of the number of tricycles that will pass through Roxas Avenue will be between 6000 and 7000 tricycles is 0.2541 or 25.41%.

2. The annual income of the family of each student in Capiz

National High School resembles a normal distribution. The distribution has

a mean of ₱ 96,000 with a standard deviation of ₱ 9,000. The school

would like to grant scholarships to the students in the lowest 10% of the

income bracket. What is the cutoff annual income for a student to be

eligible for a scholarship?

z 1 z (^2) 𝜇 = 7410

2. The applicants to a company took a 60 item entrance exam. Their scores resemble a normal distribution with a mean of 50 and a standard deviation of 4. If the company would like to have the top 20 percent of the applicants to go on to the next phase of the application process, what should be their cut-off score in the exam?
3. The Grade 11 students of Capiz National High School took a 100 item Statistics exam. The scores fit the normal distribution with a mean of 75 and a standard deviation of 10. What is the percentage of students that got a score of 80 to 85?

REFERENCES

Tizon, Melbert B. and Mesa, Helma Y., Stat Speaks, Statistics and Probability for 21st^ Century Learners, St. Bernadette Publishing House Corporation, 2016