Applied Statistics and Probability for Engineers

6-7 PROBABILITY PLOTS 219

MIND-EXPANDING EXERCISES

IMPORTANT TERMS AND CONCEPTS

In the E-book, click on any

term or concept below to

go to that subject.

Box plot

Frequency distribution

and histogram

Median, quartiles and

percentiles

Normal probability plot

Population mean

Population standard

deviation

Population variance

Random sample

Sample mean

Sample standard

deviation

Sample variance

Stem-and-leaf diagram

Time series plots

CD MATERIAL

Exponential probability

plot

Goodness of fit

Weibull probability plot

6-87. Consider the airfoil data in Exercise 6-12.

Subtract 30 from each value and then multiply the re-

sulting quantities by 10. Now compute s^2 for the new

data. How is this quantity related to s^2 for the original

data? Explain why.

6-88. Consider the quantity. For what

value of ais this quantity minimized?

6-89 Using the results of Exercise 6-87, which of the

two quantities and

will be smaller, provided that?

6-90. Coding the Data.Let i

1, 2,... , n, where aand bare nonzero constants. Find the

relationship between and , and between and

6-91. A sample of temperature measurements in a fur-

nace yielded a sample average ( ) of 835.00 and a sam-

ple standard deviation of 10.5. Using the results from

Exercise 6-90, what are the sample average and sample

standard deviations expressed in?

6-92. Consider the sample with sample

mean and sample standard deviation s. Let

What are the values of

the sample mean and sample standard deviation of the?

6-93. An experiment to investigate the survival time

in hours of an electronic component consists of placing

the parts in a test cell and running them for 100 hours

under elevated temperature conditions. (This is called an

“accelerated” life test.) Eight components were tested

with the following resulting failure times:

75, 63, 100, 36, 51, 45, 80, 90

The observation 100indicates that the unit still func-

tioned at 100 hours. Is there any meaningful measure of

location that can be calculated for these data? What is its

numerical value?

6-94. Suppose that we have a sample x 1 , x 2 , p , xnand

we have calculated and for the sample. Now an

(n1)st observation becomes available. Let and

be the sample mean and sample variance for the

sample using all n1 observations.

(a) Show how can be computed using and xn 1.

(b) Show that

(c) Use the results of parts (a) and (b) to calculate the

new sample average and standard deviation for the

data of Exercise 6-22, when the new observation is

.

6-95. The Trimmed Mean.Suppose that the data are

arranged in increasing order, T% of the observations are

removed from each end and the sample mean of the re-

maining numbers is calculated. The resulting quantity is

called a trimmed mean.The trimmed mean generally

lies between the sample mean and the sample median

. Why?
(a) Calculate the 10% trimmed mean for the yield data
in Exercise 6-17.
(b) Calculate the 20% trimmed mean for the yield data
in Exercise 6-17 and compare it with the quantity
found in part (a).
(c) Compare the values calculated in parts (a) and (b)
with the sample mean and median for the yield
data. Is there much difference in these quantities?
Why?
6-96. The Trimmed Mean.Suppose that the sample
size nis such that the quantity nT100 is not an integer.
Develop a procedure for obtaining a trimmed mean in
this case.

x

x

x 38  64

ns^2 n 1  1 n 12 s^2 n

n 1 xn 1 xn 22

n 1

xn 1 xn

s^2 n 1

xn 1

xn s^2 n

zi

zi 1 xix (^2) s, i1, 2,... , n.
x
x 1 , x 2 ,... , xn
C
F
x y sx sy.
yiabxi,
x  
gni 1 1 xix 22 gin 1 1 xi 22
gni 1 1 xia 22
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