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

Подождите немного. Документ загружается.

216 Measurement and Data Analysis for Engineering and Science

Trial

Response Total

TABLE 6.12

Sign pattern for two factors, A and B.

These average eﬀects from four runs have signiﬁcantly greater reliability

than changes computed from three runs. In addition, an interaction may

exist between the factors. An eﬀect of the interaction can be estimated from

the four runs by taking the diﬀerences between the diagonal averages, where

+ y

) −

+ y

) =

[(y

− y

) + (y

− y

)] . (6.47)

Finally, a measure of the overall level of the process can be based on the

average as

ME =

+ y

] . (6.48)

Equations 6.45 through 6.48 provide the basic structure of a factorial design

with two levels per factor and four runs. Examine the forms of the above

main eﬀect equations.

If the parentheses in Equations 6.45 through 6.47 are dropped and the

responses are placed in the order of their subscripts, these equations become

[−y

+ y

− y

+ y

] , (6.49)

[−y

− y

+ y

] , (6.50)

and

[+y

− y

+ y

] . (6.51)

A certain pattern of plus and minus signs from left to right appears in each

equation. Table 6.12 lists the pattern of plus and minus signs in columns

under each factor, A and B, the interaction, AB, and the overall gain, M.

Note that the signs of the interaction are products of the signs of the factors.

A full set of trials often is repeated to permit estimation of the eﬀects

of the inﬂuence of uncontrolled variables. The above equations can be ex-

pressed more conveniently in terms of the sums or totals of the responses

rather than the responses themselves. That is, for r runs or full sets of trials,

the total, T

, for the responses, y

, is given by adding the r responses as

Statistics 217

i=1

. (6.52)

Experiments organized as above are referred to as 2

designs, which yield

four trials for two levels and two factors. A general form of Equations 6.49

through 6.51 that provides the estimates of the eﬀects for k factors is [14]

k−1





i=1

±T





, (6.53)

where j = A, B, AB, ... . For two factors, the proper signs for each term

from left to right in Equation 6.53 are those signs in the column under the

j -th factor of Table 6.12. For example, for r = 1, y

= T

, if the main eﬀect

of factor B is to be estimated, then Equation 6.53 gives

[−T

− T

+ T

] , (6.54)

where the signs are those in the column under B in Table 6.12. For values

of k ≥ 2, a listing of the sequence of signs is given in statistics texts (for

example, see [15]). Note that Equation 6.53 must be modiﬁed to calculate

the overall mean, M, of the responses, where,

M =





i=1





, (6.55)

in which j = A, B, AB, ..., and all the signs are plus signs.

The sensitivity of a process to a factor level change generally diﬀers from

factor to factor. A small change in one factor may cause a large change in

the response while another does not. Sensitivity is the slope of the response

curve. In factorial design, the sensitivity, ζ, of the response to a certain

factor is the main eﬀect divided by the change in the factor. For example,

in the case of factor B,

− B

−

. (6.56)

A problem can arise with the application of Equation 6.56 when the factors

are attributes rather than numeric. As a result, sensitivity is usually viewed

as being determined directly by the main eﬀects themselves.

The example presented in Section 6.9 now can be analyzed using factorial

analysis. Suppose that two runs were conducted, giving the results that are

presented in Table 6.13. With this information, the main eﬀects can be

computed using Equation 6.53 and using the sign patterns in Table 6.12.

For example, the main eﬀect of factor A, detector response time, using

Equation 6.53, is

218 Measurement and Data Analysis for Engineering and Science

A: low

A: high

B: low

= 40.9

= 47.8

= 41.6

= 39.9

= 82.5

= 87.7

B: high

= 42.4

= 50.2

= 42.0

= 46.5

= 84.4

= 96.7

TABLE 6.13

Percentage changes for four trials, each conducted twice.

[−82.5 + 87.7 − 84.4 + 96.7] = 4.4. (6.57)

Similarly, M E

= 2.7 and ME

= 1.8. These results can be interpreted.

When going from the low to high level of factor A, that is, switching the

detector position from 5 mm to 10 mm, there is a 4.4 % increase in the

detector’s amplitude. Similarly, when going from the low to high level of

factor B, that is, changing from a 20 ms response detector to a 50 ms

response detector, there is a 2.7 % increase in amplitude. The interaction

main eﬀect implies that changing from the combination of a 5 mm position

of the 20 ms response detector to a 10 mm position of the 50 ms response

detector increases the amplitude by 1.8 %. Finally, the average amplitude

increase, M, is found from Equation 6.55, for r = 2 and k = 2, to be 43.9

An inherent feature of any 2

factorial design is that it is presented in a

form that can be analyzed easily using ANalysis Of VAriance (ANOVA) (for

example, see [6], [15]), provided there are two or more full sets of runs, that

is, r > 1. ANOVA yields an important piece of information. It determines

whether the eﬀects of changing the levels of factors are statistically insignif-

icant. If this is so, it means that uncontrolled variables were present in the

experiment and caused changes greater than the controlled factor changes.

If only one run is made, a 2

design provides no measure of uncontrolled

variations, known as the statistical error, and methods other than ANOVA

must be used to measure signiﬁcance.

Statistics 219

6.11 Problem Topic Summary

Topic

Review Problems

Homework Problems

Normal

1, 2, 3, 4, 6, 9, 12,

1, 2, 3, 9, 15, 17,

14, 16, 21, 25

22, 24, 25

Student’s t

5, 7, 8, 10, 11, 13, 17,

3, 4, 5, 6, 8, 11, 14,

18, 22, 23, 24

18, 19, 20, 21, 23, 24

Chi-square

15, 19, 20

6, 7, 10, 11, 12, 13,

15, 16, 25

TABLE 6.14

Chapter 6 Problem Summary

6.12 Review Problems

1. Given 1233 instantaneous pressure measurements that are distributed

normally about a mean of 20 psi with a standard deviation of 0.5 psi,

what is the probability that a measured value will be between 19 psi

and 21 psi?

2. What is the probability, in decimal form, that a normally distributed

variable will be within 1.500 standard deviations of the mean?

3. A laser pinpoints the target for an advanced aircraft weapons system.

In a system test, the aircraft simulates targeting a ﬂight-test aircraft

equipped with an optical receiver. Data recorders show that the standard

deviation of the angle of the beam trajectory is 0.1400

◦

with a mean

of 0

◦

. The uncertainty in the angle of the beam trajectory is caused

by precision errors, and the angle is distributed normally. What is the

probability, in decimal form, that the aircraft laser system will hit a

target 10 cm wide at a range of 100 m?

4. The average age of the dining clientele at a restaurant is normally dis-

tributed about a mean of 52 with a standard deviation of 20. What is

the probability, in decimal form, of ﬁnding someone between the ages of

18 and 21 in the restaurant?

5. Each of 10 engineering students measures the diameter of a spherical

ball bearing using dial calipers. The values recorded in inches are 0.2503,

220 Measurement and Data Analysis for Engineering and Science

0.2502, 0.2501, 0.2497, 0.2504, 0.2496, 0.2500, 0.2501, 0.2494, and 0.2502.

With knowledge that the diameters of the ball bearings are distributed

normally, ﬁnd the probability that the diameter of a bearing is within

+0.0005 and −0.0005 of the mean based on the sample statistics.

6. A series of acceleration measurements is normally distributed about a

mean of 5.000 m/s

with a standard deviation of 0.2000 m/s

. Find the

value such that the probability of any value occurring below that value

is 95 %.

7. Measured accelerations (in m/s

), which are normally distributed, are

9.81, 9.87, 9.85, 9.78, 9.76, 9.80, 9.89, 9.77, 9.78, and 9.85. Estimate the

range of acceleration within which the next measured acceleration would

be at 99 % conﬁdence.

8. From the data given in the previous problem, what range contains the

true mean value of the acceleration?

9. If normal distribution (population) statistics are used (incorrectly!) to

compute the sample statistics, ﬁnd the percent probability that the next

measured acceleration will be within the sample precision interval com-

puted in the previous accelerometer problem. Compute statistics as if

they were population statistics.

10. A student determines, with 95 % conﬁdence, that the true mean value

based upon a set of 61 values equals 6. The sample mean equals 4.

Determine the value of the sample standard deviation to the nearest

hundredth.

11. The values of x measured in an experiment are 5, 1, 3, and 6. Determine

with 95 % conﬁdence the upper value of the range that will contain the

next x value.

12. A student determines, with 95 % conﬁdence, that the true mean value

based upon a set of N values equals 8. The sample mean equals 7.

Assuming that N is very large (say > 100), determine the value of the

standard deviation of the means to the nearest hundredth. Remember

that the standard deviation of the means has a positive value.

13. The mean and standard deviation of a normally distributed population

are 105 and 2, respectively. Determine the percent probability that a

member of the population will have a value between 101 and 104.

14. The scores of the students who took the SAT math exam were normally

distributed with a mean of 580 and a standard deviation of 60. Determine

the percentage of students who scored greater than 750 to the nearest

hundredth of a percent.

Statistics 221

15. The percent probability that systematic eﬀects have resulted in a χ

value greater, equal to, or greater than 25 based upon 16 measurements

is (a) 5, (b) 10, (c) 90, (d) 95, or (e) 97.5.

16. Determine the percent probability that a student will score between 60

and 90 on an exam, assuming that the scores are normally distributed

with a mean of 60 and a standard deviation of 15.

17. Determine the range of scores on a test within which 95 % of 12 students

who took an exam having a mean of 60 and a standard deviation of 15,

in whole numbers.

18. Given a mean and a standard deviation of 15 and 2.0, respectively, for

a sample of 11, determine the range that contains the true variance,

estimated at 95 % conﬁdence.

19. A pressure pipeline manufacturer has performed wall thickness mea-

surements for many years and knows that the true variance of the wall

thickness of a pipe is 0.500 mm

. If the variance of a sample is 1.02 mm

ﬁnd the percent probability that this sample variance results from only

random events. Assume a sample size of 16.

20. Determine the sample skewness for the measurand values of 7, 3, 1, 5,

and 4.

21. An engineer has performed wall thickness measurements many times

and knows that the true variance of the wall thickness of a pipe is 0.500

. If the rejection criterion for sample variance is 0.7921 mm

for a

single wall, ﬁnd the probability that the rejection criterion is good for a

sample size of 21.

22. What is the probability that a student will score between 75 and 90 on

an exam, assuming that the scores are distributed normally with a mean

of 60 and a standard deviation of 15?

23. What is the probability that a student will score between 75 and 90 on

an exam, assuming that the scores are based on only three students,

with a mean of 60 and a standard deviation of 15?

24. Determine the probability that a student will score between 75 and 90 on

an examination, assuming that the scores are based upon nine students,

with a mean of 60 and a standard deviation of 15.

25. It is known that the statistics of a well-deﬁned voltage signal are given

by a mean of 8.5 V and a variance of 2.25 V

. If a single measurement of

the voltage signal is made, determine the probability that the measured

value will be between 10 V and 11.5 V.

26. What are the units of the standardized normal variate and the normal-

ized z variable?

222 Measurement and Data Analysis for Engineering and Science

6.13 Homework Problems

1. A February 14, 1997 Observer article cited a NCAA report on a fa-

mous midwestern university’s admission gap between all 1992-95 enter-

ing freshmen and the subset of entering freshman football team mem-

bers. The article reported that the mean SAT scores were 1220 for all

entering freshmen and 894 for the football team members. Assume that

the standard deviations of the SAT scores were 80 and 135 for all fresh-

men and all football team members, respectively. Determine (a) the per-

centage of all freshmen who scored greater than 1300 on their SATs, (b)

the percentage of football players who scored greater than 1300 on their

SATs, and (c) the number of football players who scored greater than

half of all of the freshman class, assuming that there were 385 football

players. State all assumptions.

2. Assume that students who took the SAT math exam were normally

distributed about a mean value of 580 with a standard deviation of 60.

Determine what percentage of the students scored higher than 750 on

the exam.

3. Using MATLAB, determine for a class of 69 the percent probability,

to four signiﬁcant ﬁgures, of getting a test score within ±1.5 standard

deviations of the mean, assuming that the test scores are distributed

according to (a) Student’s t distribution and (b) the normal distribution.

4. During an experiment, an aerospace engineering student measures a

wind tunnel’s velocity N times. The student reports the following in-

formation, based on 90 % conﬁdence, about the ﬁnite data set: mean

velocity = 25.00 m/s, velocity standard deviation = 1.50 m/s, and un-

certainty in velocity = ±2.61 m/s. Determine (a) N, (b) the standard

deviation of the means based upon this data set (in m/s), (c) the un-

certainty, at 95 % conﬁdence, in the estimate of the true mean value of

the velocity (in m/s), and (d) the interval about the sample mean over

which 50 % of the data in this set will be (in m/s).

5. An aerospace engineering student performs an experiment in a wind

tunnel to determine the lift coeﬃcient of an airfoil. The student takes

61 measurements of the vertical force using a force balance, yielding

a sample mean value of 44.20 N and a sample variance of 4.00 N

Determine (a) the percent probability that an additional measurement

will be between 45.56 N and 48.20 N, (b) the range (in N) over which the

true mean value will be, assuming 90 % conﬁdence, and (c) the range

(in N

) over which the true variance will be assuming 90 % conﬁdence.

Statistics 223

6. An airplane manufacturer intends to establish a component acceptance

criterion that is based upon sound statistical methods. Preliminary tests

on 61 acceptable components have determined that the mean load to

produce component failure is 500 psi with a standard deviation of 25

psi. Based upon this information, provide (a) an estimate, with 99 %

conﬁdence, of the value of the next (the 62nd) measured load to produce

failure, (b) an estimate, with 99 % conﬁdence, of the true mean load to

produce failure, and (c) an estimate, with 98 % conﬁdence, of the true

variance. Finally, the manufacturer wants to be 99 % conﬁdent that

if the batch sample meets the acceptance criterion. (d) Determine the

range of sample standard deviation values (in psi) that the batch sample

can have and still meet the test criterion.

7. The sample mean of 21 golf ball weights equals 0.42 N, and the sample

variance equals 0.04 N

. Determine the range (in N

) that contains the

true variance, with 90 % conﬁdence.

8. The values of x = 5, 3, 1, and 6 were measured in an experiment. Find

the range within which will contain the next data point with 95 % con-

ﬁdence.

9. The mean and standard deviation of a normally distributed population

of values x are x

= 105 and σ = 2. Find the percent probability that a

value of x will be in the range between 101 and 104.

6.4

13.6

6.4

TABLE 6.15

Expected and observed occurrences.

10. The expected number of occurrences, E

, (assuming a normal distribu-

tion) and the observed number of occurrences, O

, for 40 measurements

are given in Table 6.15. Use the χ

test to determine the probability

that the discrepancies between the observed and expected data are due

to chance alone. The choices are (a) between 95 % and 90 %, (b) be-

tween 90 % and 50 %, (c) between 50 % and 5 %, and (d) between 5 %

and 0 %.

11. For the data values of 1, 3, 5, 7, and 9, determine, with 95 % conﬁdence,

the values of the ranges that contain (a) the true mean, (b) the true

variance, and (c) the next measured value if one more data point is

taken.

12. A battery manufacturer guarantees that his batteries will last, on aver-

age, 3 years, with a standard deviation of 1 year. His claims are based

224 Measurement and Data Analysis for Engineering and Science

upon a very large population of his ‘good’ batteries. A consumer watch

group decides to test his guarantee. Their small sample of his ‘good’

batteries indicates battery lifetimes (in years) of 1.9, 2.4, 3.0, 3.5, and

4.2. Determine (a) the percent conﬁdence that the diﬀerence between

the watch group’s sample variance and manufacturer’s true variance is

due solely to random eﬀects. Next, based upon the manufacturer’s bat-

tery population average life time and standard deviation, (b) determine

the probabilities that a battery lifetime will be less than 1.9 years and

13. R measurand values have been obtained under steady-state operating

conditions. An estimate of the value of the next measurand value, the

R + 1 value, is between 2 and 12, and an estimate of the true mean

value is between 2 and 4. Both estimates are made with 90 % conﬁdence.

Determine (a) the value of R and (b) the sample variance.

14. Given that the mean and standard deviation are 10 and 1.5, respectively,

for a sample of 16, estimate with 95 % conﬁdence, the ranges within

which are (a) the true mean and (b) the true standard deviation.

15. The sample standard deviation of the length of 12 widgets taken oﬀ an

assembly line is 0.20 mm. Determine the widgets population’s standard

deviation to support the conclusion that the probability is 50 % for any

diﬀerence between the sample’s and the population’s standard deviations

to be the result of random eﬀects.

16. Determine the percent conﬁdence that an experimenter should properly

claim if the estimated true variance of a variable is between 6.16 and

14.6, based upon 31 measurements and a sample standard deviation of

17. Assuming that the performance of a class is normally distributed (which

in most cases it is not), (a) what is the probability that a student will

score above a 99 % on the ﬁnal exam if the mean is 76 % and the

standard deviation is 11 %? (b) What if the mean is only 66 % but the

standard deviation increases to 22 %?

18. The sample mean of 13 bowling balls measured from a manufacturing

line is 10.12 lbf with a sample variance of 0.28 lbf

. Determine the range

(in N) that contains the true standard deviation of all the bowling balls

assuming 90 % conﬁdence.

19. A student, working on the development of an airship, wishes to deter-

mine the quality of his pressure transducer. During a controlled air-ship

experiment, he measures the pressure (in psia) of 14.2, 14.2, 14.4, 14.8,

and 14.5. Determine the upper value of the range within the which the

next data point will be to the nearest hundredth at 95 % conﬁdence.

Statistics 225

20. Satisﬁed with the pressure transducer, an aviator takes his airship to

an assumed altitude. However, he has no way of verifying the altitude.

Therefore, he decides to measure air pressure at altitude and compare

it to that in a table. He measures pressures (in psia) of 9.8, 9.9, 10.2,

9.0, 10.4, 10.1, 10.0, and 10.6. Determine the range that contains 95 %

of the actual pressures.

21. A student determines that the true mean of a set of 31 values is 301.23

with 99 % conﬁdence, while his sample mean equaled 299.89. What is

the standard deviation of the sample? What is the standard deviation

of the means?

22. Based on a large data base, the State Highway Patrol has determined

that the average speed of Friday afternoon drivers on an interstate is

67 mph with a standard deviation of 4 mph. How many drivers of 1000

travelling on that interstate on Friday afternoon will be travelling in

excess of 72 mph?

23. A small piece of cloth was found at the scene of a crime. One suspect was

found wearing a sport coat having similar material. Ten ﬁber-diameter

measurement tests were conducted on each of the two samples. The di-

ameters (in mm) for cloth A were 3.0806, 3.0232, 2.9010, 3.1340, 3.0290,

3.1479, 3.1138, 2.9316, 2.8708, and 2.9927; for cloth B they were 2.9820,

2.9902, 3.0728, 2.9107, 2.9775, 2.9348, 2.9881, 3.2303, 2.9090, and 2.7979.

What is the percent conﬁdence that the crime-scene cloth was from the

sport coat?

24. Using the data ﬁle heights.txt that contains the heights in centimeters

of 500 college students, determine and plot the running sample mean and

running sample standard deviation of the heights, that is, the sample

mean and sample standard computed for each N (1 through 500). Also,

provide the values for N = 10, N = 100, and N = 500.

25. The following problems use the data ﬁle signal.dat that contains two

columns, each with 5000 rows of data (the ﬁrst column is the measured

velocity in m/s and the second column is the sample time in s). The

velocities were measured behind an obstruction that contained several

cables of diﬀerent diameters. The data was taken over a period of 5 s at

a sample rate of 1 kHz (1000 samples/s). Assume that the sample rate

was fast enough such that the Nyquist sampling criterion was met. The

two M-ﬁles hf.m and chinormchk.m may be useful. Do the following by

using a given M-ﬁle, writing a program, or using a spreadsheet. (a) Plot

the histogram of the velocities presented in the ﬁrst column in the data

ﬁle. Use Scott’s formula to determine the required number of bins. (b)

Plot the frequency distribution of the velocities. Use Scott’s formula.

in histogram format along with the actual number of occurrences (as