EXERCISES Section 4.6 (87-97)

87

The accompanying normal probability plot was constructed from a sample of 30 readings on tension for mesh screens behind the surface of video display tubes used in computer monitors. Does it appear plausible that the tension distribution is normal?

88

A sample of 15 female collegiate golfers was selected and the clubhead velocity $\left(\mathrm{km}/\mathrm{hr}\right)$ while swinging a driver was determined for each one, resulting in the following data (“Hip Rotational Velocities During the Full Golf Swing,” J. of Sports Science and Medicine, 2009: 296-299):

69.0	69.7	72.7	80.3	81.0
85.0	86.0	86.3	86.7	87.7
89.3	90.7	91.0	92.5	93.0

The corresponding $z$ percentiles are

-1.83	- 1.28	- 0.97	-0.73	- 0.52
-0.34	- 0.17	0.0	0.17	0.34
0.52	0.73	0.97	1.28	1.83

Construct a normal probability plot and a dotplot. Is it plausible that the population distribution is normal?

89

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

Observation	24.46	25.61	26.25	26.42	26.66
z percentile	-1.96	-1.44	-1.15	-.93	-.76
Observation	27.15	27.31	27.54	27.74	27.94
z percentile	- .60	- .45	-.32	-.19	$-.06$
Observation	27.98	28.04	28.28	28.49	28.50
z percentile	.06	.19	.32	.45	.60
Observation	28.87	29.11	29.13	29.50	30.88
z percentile	.76	.93	1.15	1.44	1.96

90

The article “A Probabilistic Model of Fracture in Concrete and Size Effects on Fracture Toughness” (Magazine of Concrete Res., 1996: 311-320) gives arguments for why fracture 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 $n=18$ observations on toughness for high-strength concrete (consistent with one of the histograms); values of $p_i=\left(i-.5\right)/18$ are also given. Construct a Weibull probability plot and comment.

Observation	.47	.58	.65	.69	.72	.74
$p_i$	.0278	.0833	.1389	.1944	.2500	.3056
Observation	.77	.79	.80	.81	.82	.84
$p_i$	.3611	.4167	.4722	.5278	.5833	.6389
Observation	.86	.89	.91	.95	1.01	1.04
$p_i$	.6944	.7500	.8056	.8611	.9167	.9722

91

Construct a normal probability plot for the fatigue-crack propagation data given in Exercise 39 (Chapter 1). Does it appear plausible that propagation life has a normal distribution? Explain.

92

The article “The Load-Life Relationship for M50 Bearings with Silicon Nitride Ceramic Balls” (Lubrication Engr., 1984: 153-159) reports the accompanying data on bearing load life (million revs.) for bearings tested at a $6.45\mathrm{kN}$ load.

47.1	68.1	68.1	90.8	103.6	106.0	115.0
126.0	146.6	229.0	240.0	240.0	278.0	278.0
289.0	289.0	367.0	385.9	392.0	505.0

a. Construct a normal probability plot. Is normality plausible?

b. Construct a Weibull probability plot. Is the Weibull distribution family plausible?

93

Construct a probability plot that will allow you to assess the plausibility of the lognormal distribution as a model for the rainfall data of Exercise 83 in Chapter 1.

94

The accompanying observations are precipitation values during March over a 30-year period in Minneapolis-St. Paul.

.77	1.20	3.00	1.62	2.81	2.48
1.74	.47	3.09	1.31	1.87	.96
.81	1.43	1.51	.32	1.18	1.89
1.20	3.37	2.10	.59	1.35	.90
1.95	2.20	.52	.81	4.75	2.05

a. Construct and interpret a normal probability plot for this data set.

b. Calculate the square root of each value and then construct a normal probability plot based on this transformed 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 probability plot of the tensile ultimate-strength data given in Exercise 13 of Chapter 1, and comment.

96

Let the ordered sample observations be denoted by $y_1,y_2,\dots ,y_n$ ( $y_1$ being the smallest and $y_n$ the largest). Our suggested check for normality is to plot the $\left({\Phi }^{-1}\left(\left(i-.5\right)/n\right),y_i\right)$ pairs. Suppose we believe that the observations come from a distribution with mean 0 , and let $w_1,\dots ,w_n$ be the ordered absolute values of the $x_i{}^{\prime }\mathrm{s}$ . A half-normal plot is a probability plot of the $w_i$ ’s. More specifically, since $P\left(\left|Z\right|\le w\right)=P\left(-w\le Z\le w\right)=$ $2\Phi \left(w\right)-1$ , a half-normal plot is a plot of the $\left({\Phi }^{-1}/\{\left[\left(i-.5\right)/n+1\right]/2\},w_i\right)$ 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: -3.78, -1.27, 1.44, -.39, 12.38, - 43.40, 1.15, - 3.96, - 2.34, 30.84.

97

The following failure time observations (1000s of hours) resulted from accelerated life testing of 16 integrated circuit chips of a certain type:

82.8, 242.0, 229.9, 11.6, 26.5, 558.9, 359.5, 244.8, 366.7, 502.5, 304.3, 204.6, 307.8, 379.1, 179.7, 212.6

Use the corresponding percentiles of the exponential distribution with $\lambda =1$ to construct a probability plot. Then explain why the plot assesses the plausibility of the sample having been generated from any exponential distribution.