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Chapter 4: Q31E (page 167)

Determine \({{\rm{z}}_{\rm{\alpha }}}\) for the following values of \({\rm{\alpha }}\): a. \({\rm{\alpha = }}{\rm{.0055}}\) b. \({\rm{\alpha = }}{\rm{.09}}\) c. \({\rm{\alpha = }}{\rm{.663}}\)

Short Answer

Expert verified

(a) The value is \({\rm{2}}{\rm{.54}}\).

(b) The value is\({\rm{1}}{\rm{.34}}\).

(c) The value is \({\rm{ - 0}}{\rm{.42}}\).

Step by step solution

01

Define variable

An unknown number, unknown value, or unknown quantity is represented by a variable, which is an alphabet or word. In the context of algebraic expressions or algebra, the variables are particularly useful.

02

Explanation

(a) For \({\rm{\alpha = 0}}{\rm{.0055}}\), \({{\rm{z}}_{\rm{\alpha }}}\) denotes the area to the right of \({{\rm{z}}_{\rm{\alpha }}}\) on a conventional normal distribution curve.

We might also say:

\(\begin{array}{c}\phi \left( {{{\rm{z}}_{\rm{\alpha }}}} \right){\rm{ = 1 - 0}}{\rm{.0055}}\\\phi \left( {{{\rm{z}}_{\rm{\alpha }}}} \right){\rm{ = 0}}{\rm{.9945}}\end{array}\)

The cdf of a regular normal distributed\({\rm{rv}}\)\({\rm{Z}}\)is\(\phi {\rm{(z)}}\).

For all\({\rm{z}}\), we check Appendix Table\({\rm{A}}{\rm{.3}}\)to see if\(\phi {\rm{(z)}}\)equals\({\rm{0}}{\rm{.9945}}\).

At the point where the row marked\({\rm{2}}{\rm{.5}}\)and the column marked\({\rm{.04}}\)cross, As, a result, the number there is\({\rm{0}}{\rm{.9945}}\).

\({{\rm{z}}_{\rm{\alpha }}}{\rm{ = 2}}{\rm{.54}}\)

Therefore, the value is \({\rm{2}}{\rm{.54}}\).

03

Explanation

(a) For \({\rm{\alpha = 0}}{\rm{.09}}\), \({{\rm{z}}_{\rm{\alpha }}}\) denotes the area to the right of \({{\rm{z}}_{\rm{\alpha }}}\) on a conventional normal distribution curve.

We might also say:

\(\begin{array}{c}\phi \left( {{{\rm{z}}_{\rm{\alpha }}}} \right){\rm{ = 1 - 0}}{\rm{.09}}\\\phi \left( {{{\rm{z}}_{\rm{\alpha }}}} \right){\rm{ = 0}}{\rm{.91}}\end{array}\)

The cdf of a regular normal distributed\({\rm{rv}}\)\({\rm{Z}}\)is\(\phi {\rm{(z)}}\).

For all\({\rm{z}}\), we check Appendix Table\({\rm{A}}{\rm{.3}}\)to see if\(\phi {\rm{(z)}}\)equals\({\rm{0}}{\rm{.91}}\).

At the point where the row marked\({\rm{1}}{\rm{.3}}\)and the column marked\({\rm{.04}}\)cross, As, a result, the number there is\({\rm{0}}{\rm{.91}}\).

\({{\rm{z}}_{\rm{\alpha }}}{\rm{ = 1}}{\rm{.34}}\)

Therefore, the value is \({\rm{1}}{\rm{.34}}\).

04

Explanation

(a) For \({\rm{\alpha = 0}}{\rm{.663}}\), \({{\rm{z}}_{\rm{\alpha }}}\) denotes the area to the right of \({{\rm{z}}_{\rm{\alpha }}}\) on a conventional normal distribution curve.

We might also say:

\(\begin{array}{c}\phi \left( {{{\rm{z}}_{\rm{\alpha }}}} \right){\rm{ = 1 - 0}}{\rm{.663}}\\\phi \left( {{{\rm{z}}_{\rm{\alpha }}}} \right){\rm{ = 0}}{\rm{.337}}\end{array}\)

The cdf of a regular normal distributed\({\rm{rv}}\)\({\rm{Z}}\)is\(\phi {\rm{(z)}}\).

For all\({\rm{z}}\), we check Appendix Table\({\rm{A}}{\rm{.3}}\)to see if\(\phi {\rm{(z)}}\)equals\({\rm{0}}{\rm{.337}}\).

At the point where the row marked\({\rm{ - 0}}{\rm{.4}}\)and the column marked\({\rm{.02}}\)cross, As, a result, the number there is\({\rm{0}}{\rm{.3372}}\).

\({{\rm{z}}_{\rm{\alpha }}}{\rm{ = - 0}}{\rm{.42}}\)

Therefore, the value is\({\rm{ - 0}}{\rm{.42}}\).

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