Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.1. Sampling and Statistics 235

lines(density(sulfurdioxide))

y=dnorm(sulfurdioxide,53.91667,10.07371);lines(y~sulfurdioxide,lty=2)

The normal density plot seems to be a poor fit.

Histogram of sulfurdioxide

Sulfurdioxide

30 40 50 60 70

0.00

0.01

0.02

0.03

0.04

Figure 4.1.3:Histogram of the sulfur dioxide concentrations in a damaged Bavar-

ian forest overlaid with a density estimate (solid line) and a normal pdf (dashed

line) with mean and variance replaced by the sample mean and standard deviations,

respectively. Data are given in Example 4.1.3.

EXERCISES

4.1.1.Twenty motors were put on test under a high-temperature setting. The
lifetimes in hours of the motors under these conditions are given below. Also, the
data are in the filelifetimemotor.rdaat the site listed in the Preface. Suppose
we assume that the lifetime of a motor under these conditions,X,hasaΓ(1,θ)
distribution.

14521222840425153

58 67 95 124 124 160 202 260 303 363

(a)Obtain a histogram of the data and overlay it with a density estimate, using

the codehist(x,pr=T); lines(density(x))where the R vectorxcontains

the data. Based on this plot, do you think that the Γ(1,θ) model is credible?

(b)Assuming a Γ(1,θ) model, obtain the maximum likelihood estimateθ̂ofθand

locate it on your histogram. Next overlay the pdf of a Γ(1,̂θ) distribution on