Chapter 4, Problem 4.5P

(a)

Interpretation Introduction

Interpretation:

Using figure 4-1, the fraction of bulbs that is expected to have a lifetime greater than $1005.3\text{ h}$ 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:

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

To Calculate: The fraction of bulbs that is expected to have a lifetime greater than $1005.3\text{ h}$ using figure 4-1

(b)

Interpretation Introduction

Interpretation:

The fraction of bulb that is expected to have a lifetime between $798.1\text{ and 901}\text{.7 h}$ 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:

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

To Calculate: The fraction of bulb that is expected to have a lifetime between $798.1\text{ and 901}\text{.7 h}$

(c)

Interpretation Introduction

Interpretation:

Using Excel NORMDIST function, the fraction of bulb that is expected to have a lifetime between $800\text{ and 900 h}$ 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:

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

To Calculate: The fraction of bulb that is expected to have a lifetime between $800\text{ and 900 h}$ using Excel NORMDIST function