Chapter 4, Problem 4.6P

(a)

Interpretation Introduction

Interpretation:

The fraction of people with tumours having a false negative indication of cancer has to be given.

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 Give: The fraction of people with tumours having a false negative indication of cancer.

(b)

Interpretation Introduction

Interpretation:

Using excel NORMDIST function, the fraction of healthy people having false positive indication of cancer has to be given.

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 Give: The fraction of healthy people having false positive indication of cancer using excel NORMDIST function.

(c)

Interpretation Introduction

Interpretation:

From spreadsheet of part (b) and with the information that $75\%$ of patients with tumours have $\text{K}$ value above selected distribution coefficient, the fraction of healthy people having a false positive result indicating tumour has to be given.

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:

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