31 The accompanying observations are on lifetime (in hours) of power apparatus insu-

Example 4.31 The accompanying observations are on lifetime (in hours) of power apparatus insu-

  lation when thermal and electrical stress acceleration were fixed at particular values (“On the Estimation of Life of Power Apparatus Insulation Under Combined Electrical and Thermal Stress,” IEEE Trans. on Electrical Insulation, 1985: 70–78).

  A Weibull probability plot necessitates first computing the 5th, 15th, . . . , and 95th percentiles of the standard extreme value distribution. The (100p)th percentile satisfies

  p 5 F(h(p)) 5 1 2 e 2e h(p)

  from which h( p) 5 ln[2ln(1 2 p)] .

  ln(x)

  ln(x)

  The pairs (22.97, 5.64), (21.82, 6.22), c, (1.10, 7.67) are plotted as points in Figure 4.38. The straightness of the plot argues strongly for using the Weibull dis- tribution as a model for insulation life, a conclusion also reached by the author of the cited article.

  ln(x) 8

  A Weibull probability plot of the insulation lifetime data

  ■

  The gamma distribution is an example of a family involving a shape parame-

  ter for which there is no transformation h( ) such that h(X) has a distribution that

  depends only on location and scale parameters. Construction of a probability plot necessitates first estimating the shape parameter from sample data (some methods for doing this are described in Chapter 6). Sometimes an investigator wishes to know

  4.6 Probability Plots

  whether the transformed variable X u has a normal distribution for some value of u

  (by convention, u50 is identified with the logarithmic transformation, in which case X has a lognormal distribution). The book Graphical Methods for Data Analysis, listed in the Chapter 1 bibliography, discusses this type of problem as well as other refinements of probability plotting. Fortunately, the wide availability of var- ious probability plots with statistical software packages means that the user can often sidestep technical details.

  EXERCISES Section 4.6 (87–97)

  87. The accompanying normal probability plot was constructed

  22–26). Would you feel comfortable estimating population

  from a sample of 30 readings on tension for mesh screens

  mean thickness using a method that assumed a normal pop-

  behind the surface of video display tubes used in computer

  ulation distribution?

  monitors. Does it appear plausible that the tension distribu- tion is normal?

  90. The article “A Probabilistic Model of Fracture in Concrete and Size Effects on Fracture Toughness” (Magazine of Con- crete Res., 1996: 311–320) gives arguments for why frac-

  ture toughness in concrete specimens should have a Weibull distribution and presents several histograms of data that appear well fit by superimposed Weibull curves. Consider

  the following sample of size n 5 18 observations on tough- ness for high-strength concrete (consistent with one of the

  histograms); values of p i 5 (i 2 .5)18 are also given.

  z percentile

  88. A sample of 15 female collegiate golfers was selected and

  p i

  the clubhead velocity (kmhr) while swinging a driver was

  determined for each one, resulting in the following data

  p i

  (“Hip Rotational Velocities During the Full Golf Swing,”

  J. of Sports Science and Medicine, 2009: 296–299):

  p i

  .6944 .7500 .8056 .8611 .9167 .9722 69.0 69.7 72.7 80.3 81.0 Construct a Weibull probability plot and comment.

  85.0 86.0 86.3 86.7 87.7 91. Construct a normal probability plot for the fatigue-crack 89.3 90.7 91.0 92.5 93.0 propagation data given in Exercise 39 (Chapter 1). Does it appear plausible that propagation life has a normal distribu-

  The corresponding z percentiles are

  tion? Explain.

  92. The article “The Load-Life Relationship for M50 Bearings 21.83 1.28 0.97 0.73 0.52 with Silicon Nitride Ceramic Balls” (Lubrication Engr., 0.34 0.17 0.0 0.17 0.34 1984: 153–159) reports the accompanying data on bearing

  0.52 0.73 0.97 1.28 1.83 load life (million revs.) for bearings tested at a 6.45 kN load.

  Construct a normal probability plot and a dotplot. Is it plau-

  sible that the population distribution is normal?

  89. Construct a normal probability plot for the following sam-

  ple of observations on coating thickness for low-viscosity paint (“Achieving a Target Value for a Manufacturing

  a. Construct a normal probability plot. Is normality

  Process: A Case Study,” J. of Quality Technology, 1992:

  plausible?

  CHAPTER 4 Continuous Random Variables and Probability Distributions

  b. Construct a Weibull probability plot. Is the Weibull dis-

  suggested check for normality is to plot the

  tribution family plausible?

  ( 21 ((i 2 .5)n), y i ) pairs. Suppose we believe that the

  93. Construct a probability plot that will allow you to assess the

  observations come from a distribution with mean 0, and

  plausibility of the lognormal distribution as a model for the

  let w 1 , c, w n

  be the ordered absolute values of the x i rs .

  rainfall data of Exercise 83 in Chapter 1.

  A half-normal plot is a probability plot of the w i rs . More specifically, since P( u Z u w) 5 P(2w Z w) 5

  94. The accompanying observations are precipitation values dur-

  2 (w) 2 1 , a half-normal plot is a plot of the

  ing March over a 30-year period in Minneapolis-St. Paul.

  ( 21 5[(i 2 .5)n 1 1]26, w i ) pairs. The virtue of this plot is that small or large outliers in the original sample will now

  1.20 3.00 1.62 2.81 2.48 appear only at the upper end of the plot rather than at both

  ends. Construct a half-normal plot for the following sample

  1.18 1.89 of measurement errors, and comment:

  4.75 2.05 97. The following failure time observations (1000s of hours) resulted from accelerated life testing of 16 integrated circuit

  a. Construct and interpret a normal probability plot for this

  chips of a certain type:

  data set.

  b. Calculate the square root of each value and then con- struct a normal probability plot based on this trans-

  formed data. Does it seem plausible that the square root

  of precipitation is normally distributed?

  c. Repeat part (b) after transforming by cube roots. 95. Use a statistical software package to construct a normal

  Use the corresponding percentiles of the exponential

  probability plot of the tensile ultimate-strength data given in

  distribution with l51 to construct a probability plot.

  Exercise 13 of Chapter 1, and comment.

  Then explain why the plot assesses the plausibility of

  96. Let the ordered sample observations be denoted by

  the sample having been generated from any exponential

  y 1 ,y 2 , c, y n ( y 1 being the smallest and y n the largest). Our

  distribution.