Q164S Assume that x聽is a random varia... [FREE SOLUTION]

91影视

Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset 91影视 Explanations$ Math

13th Edition 路 888 pages

ISBN: 9780134506593

Chapter 4: Q164S (page 281)

Assume that xis a random variable best described by a uniform distribution with $c = 10$ and $d = 90$ .
a. Find $f (x)$ .
b. Find the mean and standard deviation of x.
c. Graph the probability distribution for xand locate its mean and theintervalon the graph.
d. Find $P (x 鈮� 60)$ .
e. Find $P (x 鈮� 90)$ .
f. Find $P (x 鈮� 80)$ .
g. Find $P (渭 - 蟽鈮� x 鈮� 渭 + 蟽)$ .
h. Find $P (x > 75)$ .

Short Answer

Expert verified

a. $f (x) = \frac{1}{80}; 10 鈮� x 鈮� 90$

b. The mean is $渭 = 50$ and the standard deviation is $蟽 = 23.094$ .

c. The graph is a symmetrical curve.

d. $P (x 鈮� 60) = 0.625$

e. $P (x 鈮� 90) = 0$

f. $P (x 鈮� 80) = 0.875$

g. $P (渭 - 蟽 < x < 渭 + 蟽) = 0.57735$

h. $P (x > 75) = 0.57735$

Step by step solution

Given information

X is auniformrandom variable.

$c = 10 d = 90$

Calculate f(x)

The p.d.f of X is

$f (x) = \frac{1}{d - c}; c 鈮� x 鈮� d = \frac{1}{90 - 10} = \frac{1}{80}; 10 鈮� x 鈮� 90$

Hence, $f (x) = \frac{1}{80}; 10 鈮� x 鈮� 90$ .

Calculate mean and standard deviation

Mean= $渭$

$渭 = \frac{c + d}{2} = \frac{(10 + 90)}{2} = \frac{100}{2} = 50$

Standard deviation= $蟽$

$蟽 = \frac{d - c}{\sqrt{12}} = \frac{90 - 10}{\sqrt{12}} = \frac{80}{\sqrt{12}} = 23.094$

Hence, the mean is $渭 = 50$ and standard deviation is $蟽 = 23.094$ .

Plot the graph

$渭卤 2 蟽 = 50 卤 (2 脳 23.094) 50 卤 46.188 = (3.812, 96.188)$

Calculate P(x≤60)

$P (x 鈮� 60) = {鈭�}_{10}^{60} f (x) d x = {鈭�}_{10}^{60} \frac{1}{80} d x = \frac{1}{80} {鈭�}_{10}^{60} d x = \frac{1}{80} 脳 50 = \frac{5}{8} = 0.625$

Hence, $P (x 鈮� 60) = 0.625$ .

Calculate P(x≥90)

$P (x 鈮� 90) = 1 - P (x < 90) = 1 - {鈭�}_{10}^{90} f (x) d x = 1 - {鈭�}_{10}^{90} \frac{1}{80} d x = 1 - (\frac{1}{80} 脳 80) = 1 - 1 = 0$

Hence, $P (x 鈮� 90) = 0$ .

Calculate P(x≤80)

$P (x 鈮� 80) = 1 - {鈭�}_{80}^{90} f (x) d x = 1 - {鈭�}_{80}^{90} \frac{1}{80} d x = 1 - (\frac{1}{80} 脳 10) = 1 - \frac{1}{8} = 0.875$

Hence, $P (x 鈮� 80) = 0.875$ .

Calculate P(μ-σ≤x≤μ+σ)

$P (渭 - 蟽 < x < 渭 + 蟽) = {鈭�}_{渭 - 蟽}^{渭 + 蟽} f (x) d x = {鈭�}_{26.906}^{73.094} \frac{1}{80} d x = \frac{1}{80} 脳 46.188 = 0.57735$

Hence,role="math" localid="1660281305747" $P (渭 - 蟽 < x < 渭 + 蟽) = 0.57735$ .

Calculate

$P (x > 75) = {鈭�}_{75}^{90} f (x) d x = {鈭�}_{75}^{90} \frac{1}{80} d x = \frac{1}{80} 脳 15 = \frac{3}{16} = 0.57735$

Hence, $P (x > 75) = 0.57735$ .

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Short Answer

Step by step solution

Given information

Calculate f(x)

Calculate mean and standard deviation

Plot the graph

Calculate P(x≤60)

Calculate P(x≥90)

Calculate P(x≤80)

Calculate P(μ-σ≤x≤μ+σ)

Calculate

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