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Q. 77

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Introductory Statistics

Found in: Page 353

Book edition OER 2018

Author(s) Barbara Illowsky, Susan Dean

Pages 902 pages

ISBN 9781938168208

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Short Answer

A subway train arrives every eight minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution. a. Define the random variable. X = ___ b. X ~ _ c. Graph the probability distribution. d. f(x) = _ e. μ = _ f. σ = _____ g. Find the probability that the commuter waits less than one minute. h. Find the probability that the commuter waits between three and four minutes. i. Sixty percent of commuters wait more than how long for the train? State this in a probability question, similarly to parts g and h, draw the picture, and find the probabilit

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Measurement of variables

a.

As per basis of provided information , X is the time length commuter that wait for a train

b.

Uniform distribution of random variable X is

$X = U (0, 8)$

c. The probability distribution is

_{$\begin{array}{rcl} f (x) & = & \frac{1}{b - a} \\ = & \frac{1}{8 - 0} \\ = & \frac{1}{8} \end{array}$}

So, the graph is

d.

The calculation of part c is

$f (x) = \{\begin{cases} \frac{1}{8} \\ 0 \end{cases} 0 < x < 8 o i s f o r o t h e r w i s e$

e.

The mean value is

$\begin{array}{rcl} μ & = & \frac{a + b}{2} \\ = & \frac{0 + 18}{2} \\ = & 8 / 2 \\ = & 4 \end{array}$

f.

The value of standard deviation

$\begin{array}{rcl} σ & = & \frac{\sqrt{{(b - a)}^{2}}}{12} \\ σ & = & \frac{\sqrt{{(8 - 0)}^{2}}}{12} \\ = & 2.31 \end{array}$

g.

$P (x < 1) = b a s e \times h e i g h t = (1 - 0) \times \frac{1}{8} = 0.125$

Step 2: Calculation of variables

h. The calculation

$P (3 < x < 4) = b a s e \times h e i g h t \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} (4 - 3) \times \frac{1}{8} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 0 \end{array} \begin{array}{rcl} . \end{array} \begin{array}{rcl} 125 \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array}$

i. The calculation of probability

$\begin{array}{rcl} P (x > k) = b a s e \times h e i g h t \\ 0.60 & = & (8 - k) \times \frac{1}{8} \\ k & = & 3.2 \end{array}$

The curve of the probability

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