91Ó°ÊÓ

The Gaussian curve is given by the formula:

$y=\frac{1}{\mathrm{σ}\sqrt{2\mathrm{π}}}{e}^{-{\left(x-\mathrm{μ}\right)}^{2}2{\mathrm{σ}}^2}$

Where,

$\mathrm{μ}$is approximated by$\stackrel{¯}{x}$

$\mathrm{σ}$is approximated by s

e is the base of the natural logarithm

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

The variations from the mean value are stated in z multiples of the standard deviation, as follows:

$z=\frac{x-\mathrm{μ}}{\mathrm{σ}}≈\frac{x-\stackrel{¯}{x}}{s}$

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

(a)

The percentage of brakes that are projected to be 80% worn:

Between$x=-∞$and$x=-45800$miles, we must calculate the proportion of the area of the Gaussian curve. This may be deduced from the spreadsheet shown in figurebelow.

The formula used to calculate area is $=NORMDIST\left(45800,B2,B2,TRUE\right)\text{'}\text{'}$

The fraction of brakes expected to be 80% worn in less than 45800 miles is 0.052 .

(b)

Fraction of brakes that is expected to be 80% worn:

Calculate the percentage of the Gaussian curve's area that falls between 60000 and 70000 miles. This may be deduced from the spreadsheet shown in figure below.

The proportion of brakes projected to be 80% worn at a range between 60000 and 70000 miles is 0.361 , according to the spreadsheet.