Patrick F. Dunn, Measurement, Data Analysis, and Sensor Fundamentals for Engineering and Science, 2nd Edition

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176 Measurement and Data Analysis for Engineering and Science

FIGURE 5.18

A triangular probability density function.

3. A sports bar hosts a gaming night where students play casino games

using play money. A business major has $1500 in play money and decides

to test a strategy on the roulette wheel. The minimum bet is $100 with

no maximum. He decides to bet that the ball will land on red each time

the wheel is spun. On the ﬁrst bet, he bets the minimum. For each

consecutive spin of the wheel, he doubles his previous bet. He decides

beforehand that he will play roulette the exact number of times that his

cash stock would allow if he lost each time consecutively. What is the

probability that he will run out of money before leaving the table?

4. An engineering student samples the wall pressure exerted by a steady-

state ﬂow through a pipe 1233 times using an analog-to-digital con-

verter. Using the recommendations made in this chapter, how many

equal-interval bins should the student use to create a histogram of the

measurements? Respond to the nearest whole bin.

5. Given the probability density function pictured in Figure 5.18, compute

the height, h, that conserves the zeroth central moment.

6. Compute the ﬁrst central moment from the probability density function

pictured in Figure 5.18.

7. Compute the kurtosis of the probability density function pictured in

Figure 5.18.

8. Compute the skewness of the probability density function pictured in

Figure 5.18.

Probability 177

9. Compute the standard deviation of the probability density function pic-

tured in Figure 5.18.

10. A diagnostic test is designed to detect a cancer precursor enzyme that

exists in 1 of every 1000 people. The test falsely identiﬁes the presence

of the enzyme in 20 out of 1000 people who actually do not have the

enzyme. What is the percent chance that a person identiﬁed as having

the enzyme actually does have the enzyme?

11. What is the chance that you will throw either a 3 or a 5 on the toss of

a fair die? (a) 1/12, (b) 1/6, (c) 1/3, (d) 1/2, (e) 1/250.

12. A pressure transducer’s output is in units of volts. N samples of its

signal are taken each second. The frequency density distribution of the

sampled data has what units? (a) 1/volts, (b) volts times seconds, (c)

volts/N, (d) none; it is nondimensional, (e) seconds.

13. What is the kurtosis? (a) Bad breath, (b) the fourth central moment,

secret football play that hopefully will make a diﬀerence next season,

(e) the square of the standard deviation.

14. How many license plates showing ﬁve symbols, speciﬁcally, two letters

followed by three digits, could be made?

15. A box contains ten screws, and three of them are defective. Two screws

are drawn at random. Find the probability that neither of the two screws

is defective. Determine the probability with and without replacement.

16. The actual probability of a college student having bronchitis is 50 %. The

student health center’s diagnostic test for bronchitis has an accuracy of

80 %. Determine the percent probability that a student who has tested

positive for bronchitis actually has bronchitis.

17. A newly developed diagnostic test indicates that 80 % of students pre-

dicted to score 100 % on an exam actually do. The diagnostic test has an

accuracy of 90 %. Determine the actual probability of a student scoring

100 % on an exam.

18. Four strain gages are place on a beam to determine an unknown force,

and they are arranged in a Wheatstone bridge conﬁguration so that the

output signal is in millivolts. If N samples are recorded each second,

what are the units of the corresponding frequency density distribution?

178 Measurement and Data Analysis for Engineering and Science

FIGURE 5.19

A probability density function.

5.12 Homework Problems

1. Determine (a) the percent probability that at least 2 out of 19 students

in a classroom will have a birthday on the same birth date of the year,

(b) how many people would have to be in the room in order to have

a greater-than-50 % chance to have a birthday on the same birth date

of the year, and (c) the percent probability that at least 2 of the 19

students will have a birthday on a speciﬁc birth date of the year.

2. A cab was involved in a hit-and-run accident during the night near a

famous mid-western university. Two cab companies, the Blue and the

Gold, operate in the city near the campus. There are two facts: (1) 85

% of the cabs in the city are Gold and 15 % are Blue, and (2) a witness

identiﬁed the cab as Blue. The court tested the reliability of the witness

under the same circumstances that existed the night of the accident and

concluded that the witness correctly identiﬁed each of the two colors

80 % of the time and failed to do so 20 % of the time. Determine the

percent probability that the cab involved in the accident was Blue.

Probability 179

3. A diagnostic test is designed to detect a bad aircraft component whose

prevalence is one in a thousand. The test has a false positive rate of 5

%, where it identiﬁes a good component as bad 5 % of the time. What

is the percent chance that a component identiﬁed as bad really is bad?

4. Use the data ﬁle diam.dat. This text ﬁle contains two columns of time

(s) and diameter (µm) data in approximately 2500 rows. For the diame-

ter data only (column 2), using MATLAB, plot (a) its histogram and (b)

its frequency distribution. Use Sturgis’s formula for the number of bins,

K, as related to the number of data points, N: K = 1 + 3.322 log

HINT: MATLAB’s function hist(x,k) plots the histogram of x with k

bins. The statement [a,b] = hist(x,k) produces the column matrices

a and b, where a contains the counts in each bin and b is the center

location coordinate of each bin. MATLAB’s function bar(b,a/N) will

plot the frequency distribution, where N is the total number of x values.

5. Using the graph of the probability density function of an underlying pop-

ulation presented in Figure 5.19, determine (a) the percent probability

that one randomly selected value from this population will be between

the values of 2 and 5. If a sample of 20 values are drawn randomly from

this population, determine (b) how many will have values greater than

2 and (c) how many will have values greater than 5.

6. Let ζ be described by the probability density function p(x) = 0.75(1 −

), if (−1 ≤ x ≤ 1) and zero otherwise. Find (a) the probability dis-

tribution function, P (x), (b) the probability P r(−1/2 ≤ ζ ≤ 1/2) and

P r(1/4 ≤ ζ ≤ 2), and (c) the value of x such that P r(ζ ≤ x) = 0.95.

7. For the measurand values of 7, 3, 1, 5, and 4, determine (a) the sample

mean, (b) the sample variance, and (c) the sample skewness.

8. For the probability density function of something, p(x), shown in Figure

5.20, determine (a) P r[2 ≤ x ≤ 3], (b) P r[x ≤ 3], and (c) P r[x ≤ 7].

9. When a very high voltage is applied between two electrodes in a gas,

the gas will break down and sparks will form (much like lightning).

The voltage at which these sparks form depends on a number of vari-

ables including gas composition, pressure, temperature, humidity, and

the surface of the electrodes. In an experiment, the breakdown voltage

was measured 10 times in atmospheric air and the breakdown voltages

in units of volts were 2305, 2438, 2715, 2354, 2301, 2435, 2512, 2621,

2139, and 2239. From these measured voltages, determine the (a) mean,

(b) variance, and (c) standard deviation.

10. Assume that 14 % of the handguns manufactured throughout the world

are 8-mm handguns. A witness at the scene of a robbery states the

perpetrator was using an 8-mm handgun. The court performs a weapon

identiﬁcation test on the witness and ﬁnds that she can identify the

180 Measurement and Data Analysis for Engineering and Science

p(x)

FIGURE 5.20

A probability density function of something.

weapon correctly 73 % of the time. (a) What is the probability that an

8-mm handgun was used in the crime? (b) What is the probability if

the witness was able to identify an 8-mm handgun correctly 90 % of the

time?

Bibliography

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New York: John Wiley and Sons.

[2] I. Stewart. 1998. What a Coincidence! Scientiﬁc American June: 95-96.

[3] W.A. Rosenkrantz. 1997. Introduction to Probability and Statistics for

Scientists and Engineers. New York: McGraw-Hill.

[4] Montgomery, D.C. and G.C. Runger. 1994. Applied Statistics and Prob-

ability for Engineers. New York: John Wiley and Sons.

[5] D. Salsburg. 2001. The Lady Tasting Tea: How Statistics Revolutionized

Science in the Twentieth Century. New York: W.H. Freeman and Com-

pany.

[6] Hoﬀrage, U., Lindsey, S., Hertwig, R. and G. Gigerenzer. 2000. Commu-

nicating Statistical Information. Science 290: 2261-2262.

[7] Student. 1908. The Probable Error of a Mean. Biometrika 4: 1-25.

[8] A. Hald. 1990. A History of Probability and Statistics and Their Appli-

cation before 1750. New York: John Wiley and Sons.

[9] D.W. Scott. 1992. Multivariate Density Estimation: Theory, Practice and

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[10] H.A. Sturgis. 1926. The Choice of a Class Interval. Journal of the Amer-

ican Statistical Association 21: 65-66.

[11] Mann, H.B. and A. Wald. 1942. On the Choice of the Number of Class In-

tervals in the Application of the Chi Square Test. Annals of Mathematical

Statistics September 1942: 306-317.

[12] C.A. Williams, Jr. 1950. On the Choice of the Number and Width of

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[13] D.W. Scott. 1979. On Optimal and Data-Based Histograms. Biometrika.

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Vol. 1, Distribution Theory. 6th ed. New York: John Wiley and Sons.

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182 Measurement and Data Analysis for Engineering and Science

[15] L. Kirkup. 1994. Experimental Methods: An Introduction to the Analysis

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Statistics

CONTENTS

6.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.3 Normalized Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.4 Student’s t Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.5 Standard Deviation of the Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.6 Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.6.1 Estimating the True Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.6.2 Establishing a Rejection Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

6.6.3 Comparing Observed and Expected Distributions . . . . . . . . . . . . . . . . 206

6.7 *Pooling Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.8 *Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.9 *Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6.10 *Factorial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6.11 Problem Topic Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.12 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.13 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

That is not exactly true, but it’s probably more true than false.

Murray Winn, St. Joseph County Republican Chairman, cited in The South Bend

Tribune, South Bend, IN, November 6, 2003.

...very many have striven to discover the cause of this direction ... but they

wasted oil and labor, because, not being practical in the research of objects in

nature, being acquainted only with books, ..., they constructed certain

ratiocinations on a basis of mere opinions, and old-womanishly dreamt the

things that were not.

William Gilbert, 1600, cited in De Magnete. 1991. New York: Dover Press.

6.1 Chapter Overview

Statistics are at the heart of many claims. How many times have you heard

that one candidate is ahead of another by a certain percentage in the latest

poll or that it is safer to ﬂy than to drive? How conﬁdent can we be in

183

184 Measurement and Data Analysis for Engineering and Science

FIGURE 6.1

The probability density and distribution functions for the normal distribution.

such statements? Similar questions arise when interpreting the results of

experiments.

In this chapter we will study statistics. We start by examining some

frequently used distributions, including the normal, Student’s t, and χ

We will learn how to use them to determine the probabilities of events and

various statistical quantities. We will examine statistical inference and learn

how to estimate the characteristics of a population from ﬁnite information.

Finally, we will investigate how experiments are planned eﬃciently using

methods of statistics. After ﬁnishing with this chapter, you will have most

of the tools necessary to perform an experiment and to interpret its results

correctly.

6.2 Normal Distribution

Now, consider the normal distribution in more detail. In the limit when N

becomes very large and P is ﬁnite, assuming that the variance remains con-

stant, the binomial probability density function becomes the normal prob-

ability density function.

Statistics 185

Consider a random error to be comprised of a large number of N el-

ementary errors of equal and inﬁnitesimally small magnitude, e, with an

equally likely chance of being either positive or negative, where P = 1/2.

The normal distribution allows us to ﬁnd the probability of occurrence of

any error in the range from −Ne to +N e, where the probability density

function is

p(x) =

2πNP (1 − P )

exp



−(x − NP )

2NP (1 − P )



. (6.1)

The mean and variance are the same as the binomial distribution, NP and

NP Q, respectively, where Q = 1 −P . The higher-order central moments of

the skewness and kurtosis are 0 and 3, respectively.

Utilizing expressions for the mean, x

, and the variance, σ

, in Equa-

tion 6.1, the probability density function assumes the more familiar form

p(x) =

√

2π

exp



−

2σ

(x − x

)



. (6.2)

The normal probability density function is shown in the left plot in Fig-

ure 6.1, in which p(x) is plotted versus the nondimensional variable z =

(x − x

)/σ. Its maximum value equals 0.3989 at z = 0.

The normal probability density function is very signiﬁcant. Many proba-

bility density functions tend to the normal probability density function when

the sample size is large. This is supported by the central limit and related

theorems. The central limit theorem can be described loosely as follows [10].

Given a population of values with ﬁnite variance, if independent samples are

taken from this population, all of size N , then the new population formed by

the averages of these samples will tend to be governed by the normal proba-

bility density function, regardless of what distribution governed the original

population. Alternatively, the central limit theorem states that whatever

the distribution of the independent variables, subject to certain conditions,

the probability density function of their sum approaches the normal prob-

ability density function (with a mean equal to the sum of their means and

a variance equal to the sum of their variances) as N approaches inﬁnity.

The conditions are that (1) the variables are expressed in a standardized,

nondimensional format, (2) no single variate dominates, and (3) the sum

of the variances tends to inﬁnity as N tends to inﬁnity. The central limit

theorem also holds for certain classes of dependent random variables.

The normal probability density function describes well those situations

in which the departure of a measurand from its central tendency is brought

about by a very large number of small random eﬀects. This is most appro-

priate for experiments in which all systematic errors have been removed and

a large number of values of a measurand are acquired. This probability den-

sity function consequently has been found to be the appropriate probability

density function for many types of physical measurements. In essence, a