Answered: Given that x is a normal variable with…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Given that x is a normal variable with mean ? = 52 and standard deviation ? = 6.8, find the following probabilities. (Round your answers to four decimal places.)

(a)  P(x ≤ 60)

(b)  P(x ≥ 50)

(c)  P(50 ≤ x ≤ 60)

Expert Solution

Step 1

Given,

Mean of the normal distribution $μ = 52$

Standard deviation of the normal distribution $σ = 6.8$

Note:

$Z = \frac{X - μ}{σ} ~ N (0, 1)$

Step 2

a) $P (x \leq 60)$

The probability value is calculated as follows:

$P (x \leq 60) = P (\frac{x - μ}{σ} \leq \frac{60 - μ}{σ}) = P (Z \leq \frac{60 - 52}{6.8}) = P (Z \leq 1.1764) = ϕ (1.1764)$

where $ϕ (z)$ is the cumulative distribution of standard normal variate z.

$ϕ (1.1764)$ is calculated by using the EXCEL formula, "=NORMSDIST(1.1764)". The value obtained is 0.8802.

The value of $P (x \leq 60)$ is 0.8802.

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