Chapter 4, Problem 4.CE

(a)

Interpretation Introduction

Interpretation:

The fraction of brakes that is expected to be $80\%$ worn in less than $40860$ miles has to be calculated.

Concept Introduction:

Gaussian curve:

The Gaussian curve is given by the formula:

$y=\frac{\text{1}}{\text{σ}\sqrt{\text{2π}}}{e}^{-{(x-\text{μ)}}^{\text{2}}/{\text{2σ}}^{\text{2}}}$

Where,

$\text{μ}$ is approximated by $\overline{x}$

$\text{σ}$ is approximated by s

e is the base of the natural logarithm $(e=2.71828)$

$1/\text{σ}\sqrt{\text{2π}}$ is normalization factor.

The deviations from mean value is expressed in multiples, z, of the standard deviation and which is given as follows:

$z=\frac{x-\text{μ}}{\text{σ}}\approx \frac{x-\overline{x}}{s}$

The area under the whole curve from $z=-\infty$ to $z=+\infty$ must be unity.

To Calculate: The fraction of brakes that is expected to be $80\%$ worn in less than $40860$ miles

(b)

Interpretation Introduction

Interpretation:

The fraction that is expected to be $80\%$ worn at a mileage between $57500$ and $71020$ miles has to be calculated.

Concept Introduction:

Gaussian curve:

The Gaussian curve is given by the formula:

$y=\frac{\text{1}}{\text{σ}\sqrt{\text{2π}}}{e}^{-{(x-\text{μ)}}^{\text{2}}/{\text{2σ}}^{\text{2}}}$

Where,

$\text{μ}$ is approximated by $\overline{x}$

$\text{σ}$ is approximated by s

e is the base of the natural logarithm $(e=2.71828)$

$1/\text{σ}\sqrt{\text{2π}}$ is normalization factor.

The deviations from mean value is expressed in multiples, z, of the standard deviation and which is given as follows:

$z=\frac{x-\text{μ}}{\text{σ}}\approx \frac{x-\overline{x}}{s}$

The area under the whole curve from $z=-\infty$ to $z=+\infty$ must be unity.

To Calculate: The fraction that is expected to be $80\%$ worn at a mileage between $57500$ and $71020$ miles