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Chapter 4: Q91E (page 192)

Construct a normal probability plot for the fatigue-crack propagation data. Does it appear plausible that propagation life has a normal distribution? Explain.

Short Answer

Expert verified

Yes, a normal distribution for propagation life is conceivable.

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

Does it appear plausible that propagation life has a normal distribution

The plot of data against their respective z-percentiles is known as a normal probability plot. We must first calculate the \({{\rm{p}}_{\rm{i}}}\)values before we can calculate the percentiles.

\({{\rm{p}}_{\rm{i}}}{\rm{ = }}\frac{{{\rm{i - 0}}{\rm{.5}}}}{{\rm{n}}}\)

Where \({\rm{n}}\) denotes the total number of sample values. Then z-percentile corresponding to the \({{\rm{i}}^{{\rm{th }}}}\) sample value is the \({\left( {{\rm{100}}{{\rm{p}}_{\rm{i}}}} \right)^{{\rm{th }}}}\) percentile of standard normal distribution curve. Or in other words, they are \({\rm{z}}\)-scores corresponding to the probability \({{\rm{p}}_{\rm{i}}}\) (Use Appendix \({\rm{A - 3}}\)or use software).

The table below lists sample values and their corresponding \({\rm{p}}\) values and z-percentiles.

The normal probability curve is shown in the diagram below. The plot's trend is relatively straight, indicating that propagation life is likely to follow a normal distribution.

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Most popular questions from this chapter

Let the ordered sample observations be denoted by \({y_1}, {y_2}, \ldots , {y_n}\)(\({y_1}\) being the smallest and \({y_n}\)the largest). Our suggested check for normality is to plot the \({\phi ^{ - 1}}((i - .5)/n),{y_i})\)pairs. Suppose we believe that the observations come from a distribution with mean, and let \({w_1}, \ldots , {w_n}\)be the ordered absolute values of the \({x_i}'s\). A half 颅normal plot is a probability plot of the\({w_i}'s\). More specifically, since \(P\left( {|Z| \le w} \right) = P\left( { - w \le Z \le w} \right) = 2\phi \left( w \right) - 1\), a half-normal plot is a plot of the \(\left( {{\phi ^{ - 1}}/ \left\{ {\left( {\left( {i - .5} \right)/n + 1} \right)/2} \right\},{w_i}} \right)\)pairs. The virtue of this plot is that small or large outliers in the original sample will now appear only at the upper end of the plot rather than at both ends. Construct a half-normal plot for the following sample of measurement errors, and comment: \(23.78, 21.27,\)\(1.44, 2.39, 12.38, \)\(243.40, 1.15, 23.96, 22.34, 30.84.\)

A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of \({\rm{10}}\) per minute, whereas the second charges a flat rate of \({\rm{99}}\) for calls up to \({\rm{20}}\) minutes in duration and then \({\rm{10\% }}\)for each additional minute exceeding \({\rm{20}}\)(assume that calls lasting a non-integer number of minutes are charged proportionately to a whole-minute's charge). Suppose the consumer's distribution of call duration is exponential with parameter\({\rm{\lambda }}\).

a. Explain intuitively how the choice of calling plan should depend on what the expected call duration is.

b. Which plan is better if expected call duration is \({\rm{10}}\) minutes? \({\rm{15}}\)minutes? (Hint: Let \({{\rm{h}}_{\rm{1}}}{\rm{(x)}}\) denote the cost for the first plan when call duration is \({\rm{x}}\) minutes and let \({{\rm{h}}_{\rm{2}}}{\rm{(x)}}\)be the cost function for the second plan. Give expressions for these two cost functions, and then determine the expected cost for each plan.)

Let \({\rm{X}}\) denote the temperature at which a certain chemical reaction takes place. Suppose that \({\rm{X}}\) has pdf

\(f(x) = \left\{ {\begin{aligned}{{}{}}{\frac{1}{9}\left( {4 - {x^2}} \right)}&{ - 1拢x拢2} \\0&{{\text{ }}otherwise{\text{ }}}\end{aligned}} \right.\)

a. Sketch the graph of \({\rm{f(x)}}\).

b. Determine the cdf and sketch it.

c. Is \({\rm{0}}\) the median temperature at which the reaction takes place? If not, is the median temperature smaller or larger than\({\rm{0}}\)?

d. Suppose this reaction is independently carried out once in each of ten different labs and that the pdf of reaction time in each lab is as given. Let \({\rm{Y = }}\) the number among the ten labs at which the temperature exceeds \({\rm{1}}\). What kind of distribution does \({\rm{Y}}\) have? (Give the names and values of any parameters.)

Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with a mean of\({\bf{70}}\)and a standard deviation of\({\bf{3}}\). a. If a specimen is acceptable only if its hardness is between 67 and 75, what is the probability that a randomly chosen specimen has an acceptable hardness? b. If the acceptable range of hardness is\(\left( {{\bf{70}} - {\bf{c}},{\rm{ }}{\bf{70}} + {\bf{c}}} \right)\), for what value of c would\({\bf{95}}\% \)of all specimens have acceptable hardness? c. If the acceptable range is as in part (a) and the hardness of each of ten randomly selected specimens is independently determined, what is the expected number of acceptable specimens among the ten? d. What is the probability that at most eight of ten independently selected specimens have a hardness of less than\({\bf{73}}.{\bf{84}}\)?

If the temperature at which a certain compound melts is a random variable with mean value \({\rm{12}}{{\rm{0}}^{\rm{^\circ }}}{\rm{C}}\) and standard deviation \({{\rm{2}}^{\rm{^\circ }}}{\rm{C}}\), what are the mean temperature and standard deviation measured in \(^{\rm{^\circ }}{\rm{F}}\)? (Hint: \(^{\rm{^\circ }}{\rm{F = 1}}{\rm{.}}{{\rm{8}}^{\rm{^\circ }}}{\rm{C + 32}}{\rm{.}}\))
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