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the proportion of the total number of conceivable outcomes to the number of options in an exhaustive collection of equally likely outcomes that cause a given occurrence.

(a) P(X拢 40) denotes the probability that the yield strength is at most \({\rm{40}}\). Standardization provides:

X拢 40

only if and only if

\(\begin{array}{*{20}{c}}{\frac{{X - 43}}{{4.5}}拢\frac{{40 - 43}}{{4.5}}} \\ {\frac{{X - 43}}{{4.5}}拢\frac{{ - 3}}{{4.5}}} \\ {Z拢 - 0.67} \end{array}\)

Thus

\(P(X拢 40) = P(Z拢 - 0.67)\)

\({\rm{Z}}\)represents a standard normal distribution \(rv\)with the \({\rm{f(z)}}\) . Hence

\(P(X拢40) = P(Z拢 - 0.67) = f( - 0.67)\)

Check Appendix Table A.3 at the junction of the row marked \({\rm{ - 0}}{\rm{.67}}\) and the column marked.07 to get \({\rm{f( - 0}}{\rm{.67)}}\)

\({\rm{f( - 0}}{\rm{.67) = 0}}{\rm{.2514}}\)

Hence

P(X拢 40) = 0.2514

If \({\rm{Z}}\)is a continuous \(rv\)with the \(cdf\)\({\rm{f(z)}}\), then Then, given \({\rm{a < b}}\) for any two numbers a and b,

P(a拢 Z拢 b) = f(b) - f(a)\

\({\rm{P(X > 60)}}\)denotes the chance that the yield strength is greater than\({\rm{60}}\). Standardization provides:

\({\rm{X > 60}}\)

only if and only if

\(\begin{array}{*{20}{c}}{\frac{{{\rm{X - 43}}}}{{{\rm{4}}{\rm{.5}}}}{\rm{ > }}\frac{{{\rm{60 - 43}}}}{{{\rm{4}}{\rm{.5}}}}}\\{\frac{{{\rm{X - 43}}}}{{{\rm{4}}{\rm{.5}}}}{\rm{ > }}\frac{{{\rm{17}}}}{{{\rm{4}}{\rm{.5}}}}}\\{{\rm{Z > 3}}{\rm{.78}}}\end{array}\)

Thus

\({\rm{P(X > 60) = P(Z > 3}}{\rm{.78)}}\)

\({\rm{Z}}\)represents a standard normal distribution \(rv\)with the \({\rm{f(z)}}\) . Hence

\({\rm{P(X > 60) = P(Z > 3}}{\rm{.78) = 1 - f(3}}{\rm{.78)}}\)

\(\begin{aligned}{*{20}{c}}{{\rm{f(3}}.78) = 0{\rm{.999922}}}\\{{\rm{1 - f(3}}.78) = 1 - 0{\rm{.999922 = 0}}{\rm{.000078}}}\end{aligned}\)

Hence,

\({\rm{P(X > 60) = 0}}{\rm{.000078}}\)

(b) The yield strength value that distinguishes the strongest\({\rm{ 75\% }}\) from the rest is the \({\rm{25th}}\)percentile of the standard normal distribution, indicated as \({{\rm{z}}_{{\rm{0}}{\rm{.75}}}}\). As a reminder, \({{\rm{z}}_{\rm{\alpha }}}\)is the \({\rm{100(1 - \alpha }}{{\rm{)}}^{{\rm{th}}}}\) percentile of the standard normal distribution.

Under the typical normal distribution curve, the area to the left of \({{\rm{z}}_{{\rm{0}}{\rm{.75}}}}\) is \({\rm{0}}{\rm{.25}}\).

We might also say:

\({\rm{f}}\left( {{{\rm{z}}_{{\rm{0}}{\rm{.75}}}}} \right){\rm{ = 0}}{\rm{.25}}\)

The \(cdf\)of a typical normal distributed \(rv\)\({\rm{Z}}\)is \({\rm{f(z)}}\)

For any \({\rm{z }}\), we examine Appendix Table A.3 to see if \({\rm{f(z)}}\)equals \({\rm{0}}{\rm{.25}}\).

The two values closest to \({\rm{0}}{\rm{.25 are 0}}{\rm{.2514 and 0}}{\rm{.2483}}\), which correspond to \({\rm{z }}\)-values of \({\rm{ - 0}}{\rm{.67 and - 0}}{\rm{.68,}}\)respectively.

\(\begin{array}{*{20}{c}}{\frac{{{{\rm{z}}_{{\rm{0}}{\rm{.75}}}}{\rm{ - ( - 0}}{\rm{.68)}}}}{{{\rm{0}}{\rm{.25 - 0}}{\rm{.2483}}}}{\rm{ = }}\frac{{{\rm{ - 0}}{\rm{.67 - ( - 0}}{\rm{.68)}}}}{{{\rm{0}}{\rm{.2514 - 0}}{\rm{.2483}}}}}\\{\frac{{\left. {{{\rm{z}}_{{\rm{0}}{\rm{.75}}}}{\rm{ + 0}}{\rm{.68}}} \right)}}{{{\rm{0}}{\rm{.0017}}}}{\rm{ = }}\frac{{{\rm{0}}{\rm{.01}}}}{{{\rm{0}}{\rm{.0031}}}}}\\{{{\rm{z}}_{{\rm{0}}{\rm{.75}}}}{\rm{ = - 0}}{\rm{.68 + 0}}{\rm{.0055}}}\\{{{\rm{z}}_{{\rm{0}}{\rm{.75}}}}{\rm{ = - 0}}{\rm{.6745}}}\end{array}\)

Let \({\rm{0}}{\rm{.75}}\)be the \(rv\) value that corresponds to the z-value.

\(\begin{array}{*{20}{c}}{\frac{{{{\rm{X}}_{{\rm{0}}{\rm{.75}}}}{\rm{ - 43}}}}{{{\rm{4}}{\rm{.5}}}}}&{{\rm{ = }}{{\rm{z}}_{{\rm{0}}{\rm{.75}}}}}\\{\frac{{{{\rm{X}}_{{\rm{0}}{\rm{.75}}}}{\rm{ - 43}}}}{{{\rm{4}}{\rm{.5}}}}}&{{\rm{ = - 0}}{\rm{.6745}}}\\{{{\rm{X}}_{{\rm{0}}{\rm{.75}}}}}&{{\rm{ = 43 + (4}}{\rm{.5)( - 0}}{\rm{.6745)}}}\\{{{\rm{X}}_{{\rm{0}}{\rm{.75}}}}}&{{\rm{ = 39}}{\rm{.9648}}}\end{array}\)

As a result, the yield strength value that distinguishes the best\({\rm{ 75\% }}\)from the rest is \({\rm{39}}{\rm{.9648}}\).