Vidakovic B. Statistics for Bioengineering Sciences: With Matlab and WinBugs Support

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7.8 Exercises 267

Suppose X

, X

,... , X

are observations from this distribution. It is known

that

1−p

. What would you report as the moment-matching estimator

if the sample X

=2, X

=6, X

=1 were observed?

What is the MLE for p?

7.7. Weibull Distribution. The two-parameter Weibull distribution is given

by the density

f (x)

=aλ

a−1

−(λx)

, a >0,λ >0, x ≥0,

with mean and variance

EX =

Γ(1 +1/a)

, and Var X =

Γ(1 +2/a) −Γ(1 +1/a)

Assume that the “shape” parameter a is known and equal to 1/2.

(a) Propose two moment-matching estimators for

λ.

(b) If X

=1, X

=3, X

=2, what are the values of the estimator?

Hint:

Recall that Γ(n) =(n −1)!

7.8. Rate Parameter of Gamma. Let X

,... , X

be a sample from a gamma

distribution given by the density

f (x)

a−1

Γ(a)

−λx

, a >0,λ >0, x ≥0,

where shape parameter a is known and rate parameter

λ is unknown and

of interest.

(a) Find the MLE of

λ.

(b) Using the fact that X

+ X

+···+ X

is also gamma distributed with

parameters na and

λ, ﬁnd the expected value of MLE from (a) and show

that it is a biased estimator of

λ.

the modiﬁed estimator.

7.9. Estimating the Parameter of a Rayleigh Distribution. If two random

variables X and Y are independent of each other and normally distributed

with variances equal to

, then the variable R =

follows the

Rayleigh distribution with scale parameter

σ. An example of such a vari-

able would be the distance of darts from the target in a dart-throwing game

where the deviations in the two dimensions of the target plane are inde-

pendent and normally distributed. The Rayleigh random variable R has a

density

268 7 Point and Interval Estimators

f (r) =

exp

−

2σ

, r ≥0,

ER = σ

=2σ

(a) Find the two moment-matching estimators of

σ.

(b) Find the MLE of

σ.

=3, R

=4, R

=2, and R

=5 are Rayleigh-distributed

random observations representing the distance of a dart from the center.

Estimate the variance of the horizontal error, which is theoretically a zero-

mean normal.

(d) In Example 5.20, the distribution of a square root of an exponential

random variable with a rate parameter

λ was Rayleigh with the following

density:

f (r)

=2λr exp{−λ r

To ﬁnd the MLE for

λ, can you use the MLE for σ from (b)?

7.10. Monocytes Among Blood Cells. Eisenhart and Wilson (1943) report the

number of monocytes in 100 blood cells of a cow in 113 successive weeks.

Monocytes Frequency Monocytes Frequency

0 0 7 12

1 3 8 10

2 5 9 11

3 13 10 7

4 19 11 3

5 13 12 2

6 15 13+ 0

(a) If the underlying model is Poisson, what is the estimator of λ?

(b) If the underlying model is Binomial

B in(100, p), what is the estimator

of p?

quencies.

Hint:

Suppose the model predicts P(X = k) = p

, k = 0,1,..., 13. The ex-

pected frequency of X

= k is 113 × p

7.11. Estimation of

θ in U (0, θ). Which of the two estimators in Example 7.5 is

unbiased? Find the MSE of both estimators. Which one has a smaller MSE?

7.12. Estimating the Rate Parameter in a Double Exponential Distribu-

tion. Let X

,... , X

follow double exponential distribution with density

f (x

|θ) =

−θ|x|

, −∞< x <∞, θ >0.

7.8 Exercises 269

For this distribution, EX = 0 and Var (X ) = EX

= 2/θ

. The double expo-

nential distribution, also known as Laplace’s distribution, is a ubiquitous

model in statistics. For example, it models the difference of two exponential

variates, or absolute value of an exponential random variable, etc.

(a) Find a moment-matching estimator for

θ.

(b) Find the MLE of

θ.

=−2, X

3, X

=2 and X

=−1.

7.13. Reaction Times I. A sample of 20 students is randomly selected and

given a test to determine their reaction time in response to a given stim-

ulus. Assume that individual reaction times are normally distributed. If

the mean reaction time is determined to be

X = 0.9 (in seconds) and the

standard deviation is s

=0.12, ﬁnd

(a) The 95% conﬁdence interval for the unknown population mean

µ;

(b) The 98.5% conﬁdence interval for the unknown population mean

µ;

7.14. Reaction Times II. Under the conditions in the previous problem, assume

that the population standard deviation was known to be

σ =0.12. Find

(a) The 98.5% conﬁdence interval for the unknown mean

µ;

(b) The sample size necessary to produce a 95% conﬁdence interval for

µ of

length 0.07.

7.15. Toxins. An investigation on toxins produced by molds that infect corn

crops was performed. A biochemist prepared extracts of the mold culture

with organic solvents and then measured the amount of toxic substance

per gram of solution. From 11 preparations of the mold culture the follow-

ing measurements of the toxic substance (in milligrams) were obtained: 3,

2, 5, 3, 2, 6, 5, 4.5, 3, 3, and 4.

Compute a 99% conﬁdence interval for the mean weight of toxic substance

per gram of mold culture. State the assumption you make about the popu-

lation.

7.16. Bias of s

∗

∗. For a sample X

,... , X

from a N (µ, σ

) population, ﬁnd the

bias of s

∗

−X )

as an estimator of variance σ

7.17. COPD Patients. Acute exacerbations of disease symptoms in patients

with chronic obstructive pulmonary disease (COPD) often lead to hospital-

ization and impose a great ﬁnancial burden on the health care system. The

study by Ghanei et al. (2007) aimed to determine factors that may predict

rehospitalization in COPD patients.

A total of 157 COPD patients were randomly selected from all COPD pa-

tients admitted to the chest clinic of Baqiyatallah Hospital during the year

2006. Subjects were followed for 12 months to observe the occurrence of any

disease exacerbation that might lead to hospitalization. Over the 12-month

period, 87 patients experienced disease exacerbation. The authors found

270 7 Point and Interval Estimators

signiﬁcant associations between COPD exacerbation and monthly income,

comorbidity score, and depression using logistic regression tools. We are

not interested in these associations in this exercise, but we are interested

in the population proportion of all COPD patients that experienced disease

exacerbation over a 12-month period, p.

(a) Find an estimator of p based on the data available. What is an approxi-

mate distribution of this estimator?

(b) Find the 90% conﬁdence interval for the unknown proportion p.

conﬁdence interval as in (b) does not exceed 0.03 in length.

(d) The hospital challenges the claim by the local health system authorities

that half of the COPD patients experience disease exacerbation in a 1-year

period, claiming that the proportion is signiﬁcantly higher. Can the hospital

support their claim based on the data available? Use

α = 0.05. Would you

reverse the decision if

α were changed to 10%?

7.18. Right to Die. A Gallup Poll estimated the support among Americans for

“right to die” laws. In the survey, 1528 adults were asked whether they

favor voluntary withholding of life-support systems from the terminally ill.

The results: 1238 said YES.

(a) Find the 99% conﬁdence interval for the percentage of all adult Ameri-

cans who are in favor of “right to die” laws.

(b) If the margin of error

is to be smaller than 0.01, what sample size is

needed to achieve this requirement? Assume

=0.8.

7.19. Exponentials Parameterized by the Scale. A sample X

,... , X

was

selected from a population that has an exponential

E (λ) distribution with

a density of f (x

|λ) =

−

, x ≥0,λ >0. We are interested in estimating the

parameter

λ.

(a) What are the moment-matching and MLE estimators of

λ based on

,... , X

(b) Two independent observations Y

∼E (λ/2) and Y

∼E (2λ) are available.

Combine them (make a speciﬁc linear combination) to obtain an unbiased

estimator of

λ. What is the variance of the proposed estimator?

∼ E (1.1λ) and Z

∼ E (0.9λ) are avail-

able. An estimator of

λ in the form

λ = pZ

+(1−p)Z

, 0 ≤ p ≤1 is proposed.

What p minimizes the magnitude of bias of

λ? What p minimizes the vari-

ance of

λ?

7.20. Bias in Estimator for Exponential

λ Distribution. If the exponen-

tial distribution is parameterized with

λ as the scale parameter, f (x|λ) =

exp{−x/λ}, x ≥ 0,λ > 0, (as in MATLAB), then

λ = X is an unbiased es-

timator of

λ. However, if it is parameterized with λ as a rate parameter,

f (x

|λ) =λ exp{−λx}, x ≥0, λ >0, then

λ =1/X is biased. Find the bias of this

There are several deﬁnitions for margin of error, the most common one is half of the length

of a 95% conﬁdence interval.

7.8 Exercises 271

estimator.

Hint:

Argue that 1/

has an inverse gamma distribution

with parameters n and

λ and take the expectation.

7.21. Yucatan Miniature Pigs. Ten adult male Yucatan miniature pigs were

exposed to various durations of constant light (“Lighting”), then sacri-

ﬁced after experimentally controlled time delay (“Survival”), as described

in Dureau et al. (1996). Following the experimental protocol, entire eyes

were ﬁxed in Bouin’s ﬁxative for 3 days. The anterior segment (cornea, iris,

lens, ciliary body) was then removed and the posterior segment divided into

ﬁve regions: Posterior pole (including optic nerve head and macula) (“P”),

nasal (“N”), temporal (“T”), superior (“S”), and inferior (“I”). Specimens were

washed for 2 days, embedded in parafﬁn, and subjected to microtomy per-

pendicular to the retinal surface. Every 200

µm, a 10-µm-thick section was

selected and 20 sections were kept for each retinal region. Sections were

stained with hematoxylin. The outer nuclear layer (ONL) thickness was

measured by an image-analyzing system (Biocom, Les Ulis, France), and

three measures were performed for each section at regularly spaced inter-

vals, so that 60 measures were made for each retinal region. The experi-

mental protocol for 11 animals was as follows (Lighting and Survival times

are in weeks).

Animal Lighting duration Survival time

Control 0 0

1 1 12

2 2 10

3 4 0

4 4 4

5 4 6

6 8 0

7 8 4

8 8 8

9 12 0

10 12 4

The data set pigs.mat contains the data structure pigs with

Observe the data

pigs.pc and argue that it deviates from normality by us-

ing MATLAB’s

qqplot. Transform pigs.pc as x = (pigs.pc - 14)/(33 -14),

to conﬁne x between 0 and 1 and assume a beta

B e(a, a) distribution. The

MLE for a is complex (involves a numerical solution of equations with

digamma functions) but the moment matching estimator is straightfor-

ward.

Find a moment-matching estimator for a.

pigs.pc, pigs.p1,...,pigs.p10, representing the posterior pole measure-

ments for the 11 animals. This data set and complete data

yucatanpigs.dat

can be found on the book’s Web site page.

272 7 Point and Interval Estimators

7.22. Computer Games. According to Hamilton (1990), certain computer games

are thought to improve spatial skills. A mental rotations test, measuring

spatial skills, was administered to a sample of school children after they

had played one of two types of computer game. Construct 95% conﬁdence

intervals based on the following mean scores, assuming that the children

were selected randomly and that the mental rotations test scores had a

normal distribution in the population.

(a) After playing the “Factory” computer game:

X =22.47, s =9.44, n =19.

(b) After playing the “Stellar” computer game:

X =22.68, s =8.37, n =19.

X =18.63, s =11.13, n =

19.

7.23. Effectiveness in Treating Cerebral Vasospasm. In a study on the ef-

fectiveness of hyperdynamic therapy in treating cerebral vasospasm, Pritz

et al. (1996) reported on the therapy where success was deﬁned as clinical

improvement in terms of neurological deﬁcits. The study reported 16 suc-

cesses out of 17 patients. Using the methods discussed in the text ﬁnd 95%

conﬁdence intervals for the success rate. Does any of the methods produce

an upper bound larger than 1?

7.24. Alcoholism and the Blyth–Still Conﬁdence Interval. Genetic mark-

ers were observed for a group of 50 Caucasian alcoholics in a study that

aimed at determining whether alcoholism has (in part) a genetic basis. The

antigen (marker) B15 was present in 5 alcoholics. Find the Blyth–Still 99%

conﬁdence interval for the proportion of Caucasian alcoholics having this

antigen.

If either p or q is close to 0, then a precise (1

−α)100% conﬁdence in-

terval for the unknown proportion p was proposed by Blyth and Still

(1983). For X

∼B in(n, p),







(X −0.5) +

−α/2

−z

1−α/2

(X −0.5) −

(X −0.5)

−α/2

n +z

−α/2

+0.5) +

−α/2

1−α/2

(X +0.5) −

(X +0.5)

−α/2

n +z

−α/2







7.25. Spores of Amanita phalloides. Exercise 2.4 provides measurements in µm

of 28 spores of the mushroom Amanita phalloides.

7.8 Exercises 273

Assuming a normality of measurements, ﬁnd

(a) A point estimator for the unknown population variance

. What is the

sampling distribution of the point estimator?

(b) A 90% conﬁdence interval for the population variance;

bound U of the 90% conﬁdence interval for the variance is at most 30%

larger than the lower bound L, that is, U/L

≤1.3.

(d) Miller (1991) showed that the coefﬁcient of variation in a normal sample

of size n has an approximately normal distribution:

approx

∼ N

n −1

¸¶

Based on this asymptotic distribution, a (1

−α)100% conﬁdence interval for

the population coefﬁcient of variation

is approximately





−z

1−α/2

n −1

1−α/2

n −1





This approximation works well if n exceeds 10 and the coefﬁcient of vari-

ation is less than 0.7. Find the 95% conﬁdence interval for the population

coefﬁcient of variation

σ/µ.

7.26. CLT-Based Conﬁdence Interval for Normal Variance. Refer to Exam-

ple 7.9. Using MATLAB, simulate a normal sample with mean 0 and vari-

ance 1 of size n

=50 and ﬁnd if a 95% conﬁdence interval for the population

variance contains a 1 (the true population variance). Check this coverage

for a standard conﬁdence interval in (7.7) and for a CLT-based interval from

Example 7.9. Repeat this simulation M

=10000 times, keeping track of the

number of successful coverages. Show that the interval (7.7) achieves the

nominal coverage, while the CLT-based interval has a smaller coverage of

about 2%. Repeat the simulation for sample sizes of n

=30 and n =200.

7.27. Stent Quality Control. A stent is a tube or mechanical scaffold used

to counteract signiﬁcant decreases in vessel or duct diameter by acutely

propping open the conduit (Fig. 7.10). Stents are often used to alleviate

diminished blood ﬂow to organs and extremities beyond an obstruction in

order to maintain an adequate delivery of oxygenated blood.

In the production of stents, the quality control procedure aims to identify

defects in composition and coating. Precision z-axis measurements (10 nm

and greater) are obtained along with surface roughness and topographic

surface ﬁnish details using a laser confocal imaging system (an example

is the Olympus LEXT OLS3000). Samples of 50 stents from a production

process are selected every hour. Typically, 1% of stents are nonconforming.

Let X be the number of stents in the sample of 50 that are nonconforming.

274 7 Point and Interval Estimators

Fig. 7.10 A stent used to restore blood ﬂow following a heart attack, in a procedure called

percutaneous coronary intervention.

A production problem is suspected if X exceeds its mean by more than three

standard deviations.

(a) Find the critical value for X that will implicate the production problem.

(b) Find an approximation for the probability that in the next-hour batch of

50 stents, the number X of nonconforming stents will be critical, i.e., will

raise suspicions that the process might have a problem.

is unknown. How would one estimate p by taking a 50-stent sample? Is the

proposed estimator unbiased?

(d) Suppose now that a batch of 50 stents produced X

= 1. Find the 95%

conﬁdence interval for p.

7.28. Right to Die. A Gallup poll was taken to estimate the support among

Americans for “right to die” laws. In the survey, 1528 adults were asked

whether they favor voluntary withholding of life-support systems from the

terminally ill; 1238 said yes. One is interested in conducting a similar study

on a university campus.

If the margin of error is to be smaller than 2%, what sample size is needed

to achieve that requirement. Assume

p = 0.8.

7.29. Clopper–Pearson and 3/n-Rule Conﬁdence Intervals. Using MAT-

LAB compare the performance of Clopper–Pearson and 3/n-rule conﬁdence

intervals when X

= 0. Use α = 0.001, 0.005,0.01,0.05,0.1 and n = 10 : 10 :

200. Which interval is superior and under what conditions?

7.30. Seventeen Pairs of Rats, Carbon Tetrachloride, and Vitamin B. In

a widely cited experiment by Sampford and Taylor (1959), 17 pairs of rats

were formed by selecting pairs from the same litter. All rats were given

carbon tetrachloride, and one rat from each pair was treated with vitamin

, while the other served as a control. In 7 of 17 pairs, the treated rat

outlived the control rat.

(a) Based on this experiment, estimate the population proportion p of pairs

in which the treated rat would outlive the control rat.

7.8 Exercises 275

(b) If the estimated proportion in (a) is the “true” population probability,

what is the chance that in an independent replication of this experiment

one will get exactly 7 pairs (out of 17) in which the treated rat outlives the

control.

contain 1/2? What does p

=1/2 mean in the context of this experiment, and

what do you conclude from the conﬁdence interval.

Would the conclusion be the same if in 140 out of 340 pairs the treated rat

outlived the control?

(d) The length of the 95% conﬁdence interval based on n

=17 in (c) may be

too large. What sample size (number of rat pairs) is needed so that the 95%

conﬁdence interval has a length of

` =0.2?

7.31. Hemocytometer Counts. A set of 1600 squares on a hemocytometer is

inspected and the number of cells is counted in each square. The number of

squares with a particular count is given in the table below:

Count 0 1 2 3 4 5 6 7

# Squares 5 24 77 139 217 262 251 210

Count 8 9 10 11 12 13 14 15

# Squares 175 108 63 36 20 9 2 1

Assume that the count has a Poisson distribution P oi(λ) and ﬁnd an esti-

mator of

λ.

7.32. Predicting Alkaline Phosphatase. Refer to BUPA liver disorder data,

BUPA.dat|mat|xlsx. The second column gives measurements of alka-

line phosphatase among 345 male individuals affected by liver disorder. If

variable X represents the logarithm of this measurement, its distribution

is symmetric and bell-shaped, so it can be assumed normal. From the data,

X =4.21 and s

=0.0676.

Suppose that a new patient with liver disorder just checked in. Find the

95% prediction interval for his log-level of alkaline phosphatase?

(a) Assume that the population variance is known and equal to 1/15.

(b) Assume that the population variance is not known.

lation mean. Why is the interval in (b) larger?

276 7 Point and Interval Estimators

MATLAB FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch7.Estim/

AmanitaCI.m, arcsinint.m, bickellehmann.m, clopperint.m,

CLTvarCI.m, confintscatterpillar.m, crouxrouss.m, crouxrouss2.m,

cyclosporine.m, dists2.m, ginimd.m, ginimd2.m, lfev.m, MaxwellMLE.m,

MixtureModelExample.m, muscaria.m, plotlike.m, Rutherford.m,

simuCI.m, tolerance.m, waldsimulation.m

neuronfires.mat

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