Answered: A population of values has a normal…

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

A population of values has a normal distribution with μ=161 and σ=16.3. You intend to draw a random sample of size n=45.

Find the probability that a sample of size n=45 is randomly selected with a mean between 155.4 and 165.6.

P(155.4 < M < 165.6) =

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Expert Solution

Step 1

The probability that a sample of size $n = 45$ is randomly selected with a mean between 155.4 and 165.6 is,

$\begin{array}{rcl} P (155.4 < M < 165.6) & = & P (\frac{155.4 - 161}{(\frac{16.3}{\sqrt{45}})} < \frac{M - μ}{(\frac{σ_{\bar{x}}}{\sqrt{n}})} < \frac{165.6 - 161}{(\frac{16.3}{\sqrt{45}})}) \\ = & P (\frac{- 5.6}{2.42986} < z < \frac{4.6}{2.42986}) \\ = & P (- 2.305 < z < 1.893) \\ = & P (z < 1.893) - P (z < - 2.305) \end{array}$

The probability of z less than –2.305 can be obtained using the excel formula “=NORM.S.DIST(–2.305,TRUE)”. The probability value is 0.01058.

The probability of z less than 1.893 can be obtained using the excel formula “=NORM.S.DIST(1.893,TRUE)”. The probability value is 0.97082.

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