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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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Most popular questions from this chapter

Let \({\rm{X}}\) denote the voltage at the output of a microphone, and suppose that \({\rm{X}}\) has a uniform distribution on the interval from \({\rm{ - 1}}\) to \({\rm{1}}\). The voltage is processed by a 鈥渉ard limiter鈥� with cut-off values \({\rm{ - }}{\rm{.5}}\) and \({\rm{.5}}\), so the limiter output is a random variable \({\rm{Y}}\) related to \({\rm{X}}\) by \({\rm{Y = X}}\) if \({\rm{|X|}} \le {\rm{.5,Y = }}{\rm{.5}}\) if \({\rm{X > }}{\rm{.5}}\), and \({\rm{Y = - }}{\rm{.5}}\) if \({\rm{X < - }}{\rm{.5}}\). a. What is \({\rm{P(Y = }}{\rm{.5)}}\)? b. Obtain the cumulative distribution function of \({\rm{Y}}\) and graph it.

A \(12\)-in. bar that is clamped at both ends is to be subjected to an increasing amount of stress until it snaps. Let Y = the distance from the left end at which the break occurs. Suppose Y has pdf

\(f\left( y \right) = \left\{ {\begin{array}{*{20}{c}}{\left( {\frac{1}{{24}}} \right)y\left( {1 - \frac{y}{{12}}} \right)\,\,\,\,\,0 \le y \le 12}\\{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise}\end{array}} \right.\)

Compute the following: a. The cdf of Y, and graph it. b.\(P\left( {Y \le 4} \right), P\left( {Y > 6} \right)\), and\(P\left( {4 \le Y \le 6} \right)\)c. E(Y), E(Y2 ), and V(Y) d. The probability that the breakpoint occurs more than \(2\;\)in. from the expected breakpoint. e. The expected length of the shorter segment when the break occurs.

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a. What is the probability that the concentration exceeds. \({\rm{50}}\)?

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\({\rm{F(x) = }}\left\{ {\begin{array}{*{20}{c}}{\rm{0}}&{{\rm{x}} \le {\rm{0}}}\\{\frac{{\rm{x}}}{{\rm{4}}}\left( {{\rm{1 + ln}}\left( {\frac{{\rm{4}}}{{\rm{x}}}} \right)} \right)}&{{\rm{0 < x}} \le {\rm{4}}}\\{\rm{1}}&{{\rm{x > 4}}}\end{array}} \right.\)

(This type of cdf is suggested in the article 鈥淰ariability in Measured Bedload Transport Rates鈥� (Water 91影视 Bull., \({\rm{1985:39 - 48}}\)) as a model for a certain hydrologic variable.) What is a. \({\rm{P(X}} \le {\rm{1)}}\)? b. \({\rm{P(1}} \le {\rm{X}} \le {\rm{3)}}\)? c. The pdf of \({\rm{X}}\)?

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