Solutions - 5180-04

(a)

(1) Write $X=\sigma Z+\mu$ and substitute into the probability:

$$\begin{array}{c}X=10Z+75\\ \\ P[70\le X\le 80]\gg \gg P[70\le 10Z+75\le 80]\gg \gg P[-\frac{1}{2}\le Z\le \frac{1}{2}]\end{array}$$

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(2) Express in terms of $\mathrm{\Phi }$ and evaluate:

$$\begin{array}{c}\mathrm{\Phi }\left(\frac{1}{2}\right)-\mathrm{\Phi }(-\frac{1}{2})\gg \gg 2\mathrm{\Phi }\left(\frac{1}{2}\right)-1\\ \\ \gg \gg 2(0.6915)-1\gg \gg 0.383\end{array}$$

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(b)

(1) Interpret problem:

Since we wish to find the top $5\%$, we wish to find $z$ such that $\mathrm{\Phi }(z)=0.95$.

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(2) Use lookup table to find $z$:

$$\mathrm{\Phi }(z)=0.95\gg \gg z\approx 1.6449$$

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(3) Solve for $X$ using $X=10Z+75$:

$$X\approx 10(1.6449)+75\gg \gg 91.449$$

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(c)

(1) Interpret problem:

Since we wish to find the 25th percentile, we wish to find $z$ such that $\mathrm{\Phi }(z)=0.25$.

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(2) Use lookup table to find $z$:

$$\mathrm{\Phi }(z)=0.25\gg \gg z\approx -0.6745$$

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(3) Solve for $X$ using $X=10Z+75$:

$$X\approx 10(-0.6745)+75\gg \gg 68.255$$