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Chapter 4: Q89E (page 192)

The accompanying sample consisting of observations \({\rm{n = 20}}\) on dielectric breakdown voltage of a piece of epoxy resin appeared in the article "Maximum Likelihood Estimation in the \({\rm{3}}\)-Parameter Weibull Distribution (IEEE Trans. on Dielectrics and Elec. Insul., \({\rm{1996}}\): \({\rm{43 - 55)}}\). The values of \({\rm{(i - }}{\rm{.5)/n}}\) for which \({\rm{z}}\) percentiles are needed are \({\rm{(1 - }}{\rm{.5)/20 = }}{\rm{.025,(2 - }}{\rm{.5)/20 = }}\)\({\rm{.075, \ldots }}\), and\({\rm{.975}}\). Would you feel comfortable estimating population mean voltage using a method that assumed a normal population distribution?

\(\begin{array}{*{20}{c}}{{\rm{ Observation }}}&{{\rm{24}}{\rm{.46}}}&{{\rm{25}}{\rm{.61}}}&{{\rm{26}}{\rm{.25}}}&{{\rm{26}}{\rm{.42}}}&{{\rm{26}}{\rm{.66}}}\\{{\rm{ zpercentile }}}&{{\rm{ - 1}}{\rm{.96}}}&{{\rm{ - 1}}{\rm{.44}}}&{{\rm{ - 1}}{\rm{.15}}}&{{\rm{ - }}{\rm{.93}}}&{{\rm{ - }}{\rm{.76}}}\\{{\rm{ Observation }}}&{{\rm{27}}{\rm{.15}}}&{{\rm{27}}{\rm{.31}}}&{{\rm{27}}{\rm{.54}}}&{{\rm{27}}{\rm{.74}}}&{{\rm{27}}{\rm{.94}}}\\{{\rm{ zpercentile }}}&{{\rm{ - }}{\rm{.60}}}&{{\rm{ - }}{\rm{.45}}}&{{\rm{ - }}{\rm{.32}}}&{{\rm{ - }}{\rm{.19}}}&{{\rm{ - }}{\rm{.06}}}\\{{\rm{ Observation }}}&{{\rm{27}}{\rm{.98}}}&{{\rm{28}}{\rm{.04}}}&{{\rm{28}}{\rm{.28}}}&{{\rm{28}}{\rm{.49}}}&{{\rm{28}}{\rm{.50}}}\\{{\rm{ zpercentile }}}&{{\rm{.06}}}&{{\rm{.19}}}&{{\rm{.32}}}&{{\rm{.45}}}&{{\rm{.60}}}\\{{\rm{ Observation }}}&{{\rm{28}}{\rm{.87}}}&{{\rm{29}}{\rm{.11}}}&{{\rm{29}}{\rm{.13}}}&{{\rm{29}}{\rm{.50}}}&{{\rm{30}}{\rm{.88}}}\\{{\rm{ zpercentile }}}&{{\rm{.76}}}&{{\rm{.93}}}&{{\rm{1}}{\rm{.15}}}&{{\rm{1}}{\rm{.44}}}&{{\rm{1}}{\rm{.96}}}\end{array}\)

Short Answer

Expert verified

Yes, we feel comfortable estimating population mean voltage using a method that assumed a normal population distribution.

Step by step solution

01

Definition

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Would you feel comfortable estimating population mean voltage

NORMAL PROBABILITY PLOT

We must generate a normal probability plot to assess if the distribution of variables is nearly normally distributed.

A scatterplot with the observations on the horizontal axis and the z-percentiles on the vertical axis is called a normal probability plot.

It is reasonable to presume that the distribution of the observations is substantially normal if the pattern in the normal probability plot is broadly linear and does not include severe curvature.

We may infer that the distribution of the observations is about normal since the produced normal probability plot has no major curvature and is roughly linear. We would feel safe calculating population mean voltage using a method that assumed a normal population distribution.

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Most popular questions from this chapter

Suppose Appendix Table \({\rm{A}}{\rm{.3}}\)contained \({\rm{\Phi (z)}}\)only for\({\rm{z}} \ge {\rm{0}}\). Explain how you could still compute

a.\({\rm{P( - 1}}{\rm{.72}} \le {\rm{Z}} \le {\rm{ - }}{\rm{.55)}}\)

b.\({\rm{P( - 1}}{\rm{.72}} \le {\rm{Z}} \le {\rm{.55)}}\)

Is it necessary to tabulate \({\rm{\Phi (z)}}\)for \({\rm{z}}\)negative? What property of the standard normal curve justifies your answer?

Evaluate the following:

a. \({\rm{\Gamma (6)}}\)

b. \({\rm{\Gamma (5/2)}}\)

c. \({\rm{F(4;5)}}\) (the incomplete gamma function) and \({\rm{F(5;4)}}\)

d. P(X拢 5)when \({\rm{X}}\) has a standard gamma distribution with\({\rm{\alpha = 7}}\).

e. \({\rm{P(3 < X < 8)}}\)when \({\rm{X}}\)has the distribution specified in (d).

As in the case of the Weibull and Gamma distributions, the lognormal distribution can be modified by the introduction of a third parameter \({\rm{\gamma }}\) such that the pdf is shifted to be positive only for \({\rm{\chi > \gamma }}\)The article cited in Exercise \({\rm{4}}{\rm{.39}}\)suggested that a shifted lognormal distribution with shift (i.e., threshold) \({\rm{ = 1}}{\rm{.0}}\), mean value \({\rm{ = 2}}{\rm{.16, }}\), and standard deviation \({\rm{ = 1}}{\rm{.03}}\) would be an appropriate model for the \({\rm{rv X = }}\) maximum-to-average depth ratio of a corrosion defect in pressurized steel.

a. What are the values of \({\rm{\mu and \sigma }}\)for the proposed distribution?

b. What is the probability that depth ratio exceeds \({\rm{2}}\)?

c. What is the median of the depth ratio distribution?

d. What is the \({\rm{99th}}\) percentile of the depth ratio distribution?

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.

Let\({\bf{X}}\)denote the data transfer time (ms) in a grid computing system (the time required for data transfer between a 鈥渨orker鈥� computer and a 鈥渕aster鈥� computer. Suppose that X has a gamma distribution with mean value\({\bf{37}}.{\bf{5}}{\rm{ }}{\bf{ms}}\)and standard deviation\({\bf{21}}.{\bf{6}}\)(suggested by the article 鈥淐omputation Time of Grid Computing with Data Transfer Times that Follow a Gamma Distribution,鈥� Proceedings of the First International Conference on Semantics, Knowledge, and Grid, 2005).

a. What are the values of\({\rm{\alpha }}\)and\({\rm{\beta }}\)?

b. What is the probability that data transfer time exceeds\({\bf{50}}{\rm{ }}{\bf{ms}}\)?

c. What is the probability that data transfer time is between\({\bf{50}}\)and\({\bf{75}}{\rm{ }}{\bf{ms}}\)?
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