This document discusses a case study on the normal probability distribution. It examines the weekly wages of 1000 workers which are normally distributed with a mean of Rs. 70 and standard deviation of Rs. 5. It estimates the number of workers whose weekly wages are: (i) Between Rs. 70 and Rs. 72 (155 workers) (ii) Between Rs. 69 and Rs. 72 (235 workers) (iii) More than Rs. 75 (159 workers) (iv) Less than Rs. 63 (81 workers). The document explains how to solve each case using the normal distribution and z-table. It concludes by asking students to explain the case study on normal probability distribution.

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Unit No.5.
PROBABILITY DISTRIBUTION
Presented By:
Dr. V. M. Tidake
Ph. D (Financial Management), MBA(FM), MBA(HRM) BE(Chem)
Dean, EDP & Associate Professor MBA
1
Sanjivani College of Engineering, Kopargaon
Department of MBA
www.sanjivanimba.org.in

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302-DECISION SCIENCE
Unit No.5. Probability
5.2.7 Case 1: Normal
Probability Distribution
Presented By:
Dr. V. M. Tidake
Ph. D (Financial Management), MBA(FM), MBA(HRM) BE(Chem)
Dean EDP & Associate Professor MBA
2
Sanjivani College of Engineering, Kopargaon
Department of MBA
www.sanjivanimba.org.in

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NETWORK ANALYSIS
 At the End of the Session Student will be able to
understand-
A. Case 1: Normal Probability Distribution

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Probability Distribution
The weekly wages of 1000 workers are normally
distributed around a mean of Rs. 70 and Standard
Deviation of Rs. 5. Estimate the number of workers whose
weekly wages will be-
i. Between Rs. 70 and Rs. 72
ii. Between Rs. 69 and Rs. 72
iii. More than Rs. 75
iv. Less than Rs. 63

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Case 1: Normal Distribution
Let x denote the weekly wages in Rupees.
Thus, x is a normal variable with mean, m=70 and 𝜎=5
Standard Normal variate corresponding to x is-
z =
𝑥 − 𝑚
𝜎
=
𝑥 − 70
5

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Case 1: Normal Distribution
i. Let us find P(wages between Rs. 70 and Rs. 72) i.e. P(70≤x≤72)
Now, when x1=70,
z1 =
70 − 70
5
= 0
And x2= 72,
z2 =
72 − 70
5
= 0.4
P(70≤x≤72)= P(0≤z≤0.4)=0.1554 -(From the z Table)
Probability that the workers will have weekly wages between Rs.
70 and Rs. 72 is 0.1554
Hence, the number of such workers among 1000 workers will be-
f(70≤x≤72)=1000* P(70≤x≤72)=1000*0.1554=155.4 i.e. 155

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Case 1: Normal Distribution
ii. Let us find P(wages between Rs. 69 and Rs. 72) i.e. P(69≤x≤72)
Now, when x1=69,
z1 =
69 − 70
5
= −0.2
And x2= 72,
z2 =
72 − 70
5
= 0.4
P(69≤x≤72)= P(-0.2≤z≤0.4)
= P(-0.2≤z≤0)+P(0≤z≤0.4)
= P(0≤z≤0.2)+P(0≤z≤0.4) - (By Symmetry)
= 0.0793+0.1554=0.2347 -(From the z Table)
Probability that the workers will have weekly wages between Rs. 69 and Rs. 72 is
0.2347
Hence, the number of such workers among 1000 workers will be-
f(69≤x≤72)=1000* P(69≤x≤72)=1000*0.2347=234.7 i.e. 235

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Case 1: Normal Distribution
iii. Let us find P(wages more than Rs. 75) i.e. P(x≥75)
Now, when x=75,
z1 =
75 − 70
5
= 1
P(x≥75) = P(z ≥ 1) = P(0≤x≤∞) - P(0≤x≤1)
= 0.5 - 0.3413 = 0.1587 -(From the z Table)
Probability that the workers will have weekly wages more
than Rs. 75 is 0.1587
Hence, the number of such workers among 1000 workers
will be-
f(x≥75)=1000* P(x≥75)=1000*0.1587=158.7 i.e. 159

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Case 1: Normal Distribution
iv. Let us find P(wages less than Rs. 63) i.e. P(x≤63)
Now, when x=63,
z1 =
63 − 70
5
= −1.4
P(x ≤ 63) = P(z ≤ -1.4) = P(z ≥ 1.4) = P(0≤z≤∞) - P(0≤z≤1.4)
= 0.5 - 0.4192 = 0.0808 -(From the z Table)
Probability that the workers will have weekly wages less
than Rs. 63 is 0.0808
Hence, the number of such workers among 1000 workers
will be-
f(x ≤ 63)=1000* P(x ≤ 63)=1000*0.0808=80.8 i.e. 81

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EXERCISE
 Explain case 1: Normal Probability Distribution

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For More Details Contact
Dr. V M Tidake
tidkevishal@gmail.com
tidkevishalmba@sanjivani.org.in