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=86.1μ=86.1 and σ=46.3σ=46.3. You intend to draw a random sample of size n=232n=232. Please answer the following questions, and show your answers to 1 decimal place.

Find the value separating the bottom 88% values from the top 12% values.

Find the sample mean separating the bottom 88% sample means from the top 12% sample means.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Expert Solution

Step 1

$\mathrm{{\color{Blue} \mathbf{Given:}}\;\mu=86.1,\;\sigma=46.3,\;n=232}$

$\mathbf{P_{88}\Rightarrow 88^{th}\;percentile}$

88^th implies area of 0.88 to the left of the normal curve.

Refer Standard normal table/Z-table, Lookup for z-score corresponding to area 0.88 to the left of the normal curve OR use excel formula "=NORM.S.INV(0.88)" to find the z-score.

$\mathbf{= P(Z < 1.175) = 0.88}$

$\mathbf{\therefore z=1.175}$

For single value:

$\mathbf{Formula}: z=\frac{x-\mu}{\sigma}$

Plug in all the values in the formula:

$1.175=\frac{x-86.1}{46.3}$

$x=86.1+(1.175*46.3)$

$x=86.1+54.4025$

$x=140.5025$

$\mathrm{Round\; the \;value \;to\; one\;decimal\; place.}$

$\mathbf{x=140.5}$

$\mathbf{\therefore P_{88} (for \;single\; values) ={\color{Red} 140.5}}$

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