Answered: Probability Question

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Expert Solution

Step 1

Given $μ = 58 a n d σ = 6$ .

a) The empirical rule states that 95% of data values fall within two standard deviations of mean, i.e.
$\begin{array}{rcl} μ \pm 2 σ & = & 58 \pm 2 (6) \\ = & 58 \pm 12 \\ = & (46, 70) \end{array}$

Therefore, 46 and 70 is the answer.

Step 2

Approximately 68% of the data values fall within one standard deviation of mean, i.e.
$\begin{array}{rcl} μ \pm σ & = & 58 \pm 6 \\ = & (52, 64) \end{array}$

Approximately 99.7% of the data values fall within three standard deviations of mean, i.e.
$\begin{array}{rcl} μ \pm 3 σ & = & 58 \pm 3 (6) \\ = & 58 \pm 18 \\ = & (40, 76) \end{array}$

Again by empirical rule,
$(\frac{68}{2}) % = 34 % d a t a l i e s w i t h i n (μ - σ, μ) a n d t h e r e s t o f 34 % d a t a l i e s w i t h i n (μ, μ + σ)$ .

Similarly,
$(\frac{95}{2}) % = 47.5 % d a t a l i e s w i t h i n (μ - 2 σ, μ) a n d t h e r e s t o f 47.5 % d a t a l i e s w i t h i n (μ, μ + 2 σ)$ .

Then, the required percentage is
$\begin{array}{rcl} r e q u i r e d p e r c e n t a g e & = & P e r c e n t a g e w h i c h f a l l s w i t h i n (μ, μ + σ) - P e r c e n t a g e w h i c h f a l l s w i t h i n (μ - σ, μ) \\ = & 47.5 - 34 \\ = & 13.5 % \end{array}$