Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences

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old are drawn from these populations, ﬁnd the probability that the difference in percent

of total natural teeth loss is less than 5 percent between the two populations.

Solution: We assume that the sampling distribution is approximately normal.

The mean difference in proportions of those losing all their teeth is

and the variance is

The area of interest under the curve of is that to the left of .05. The

corresponding z value is

Consulting Table D, we find that the area to the left of is

.2420.

■

EXERCISES

5.6.1 According to the 2000 U.S. Census Bureau (A-8), in 2000, 9.5 percent of children in the state of

Ohio were not covered by private or government health insurance. In the neighboring state of

Pennsylvania, 4.9 percent of children were not covered by health insurance. Assume that these

proportions are parameters for the child populations of the respective states. If a random sample

of size 100 children is drawn from the Ohio population, and an independent random sample of

size 120 is drawn from the Pennsylvania population, what is the probability that the samples would

yield a difference, of .09 or more?

5.6.2 In the report cited in Exercise 5.6.1 (A-8), the Census Bureau stated that for Americans in the age

group 18–24 years, 64.8 percent had private health insurance. In the age group 25–34 years, the per-

centage was 72.1. Assume that these percentages are the population parameters in those age groups

for the United States. Suppose we select a random sample of 250 Americans from the 18–24 age

group and an independent random sample of 200 Americans from the age group 25–34; ﬁnd the prob-

ability that is less than 6 percent.

5.6.3 From the results of a survey conducted by the U.S. Bureau of Labor Statistics (A-9), it was esti-

mated that 21 percent of workers employed in the Northeast participated in health care beneﬁts

programs that included vision care. The percentage in the South was 13 percent. Assume these

percentages are population parameters for the respective U.S. regions. Suppose we select a sim-

ple random sample of size 120 northeastern workers and an independent simple random sample

of 130 southern workers. What is the probability that the difference between sample proportions,

will be between .04 and .20?p

- p

z =-.70

z =

.05 - 1.082

2.00186

=-.70

- p

11 - p

1.3421.662

250

1.2621.742

200

= .00186

= .34 - .26 = .08

- p

EXERCISES

157

158 CHAPTER 5 SOME IMPORTANT SAMPLING DISTRIBUTIONS

5.7 SUMMARY

This chapter is concerned with sampling distributions. The concept of a sampling distri-

bution is introduced, and the following important sampling distributions are covered:

1. The distribution of a single sample mean.

2. The distribution of the difference between two sample means.

3. The distribution of a sample proportion.

4. The distribution of the difference between two sample proportions.

We emphasize the importance of this material and urge readers to make sure that

they understand it before proceeding to the next chapter.

SUMMARY OF FORMULAS FOR CHAPTER 5

Formula Number Name Formula

5.3.1 z-transformation

for sample mean

Z =

- m

s>1n

5.4.1 z-transformation for

difference between

two means

5.5.1 z-transformation for

sample proportion

5.5.2 Continuity correction

when x  np

5.5.3 Continuity correction

when x  np

5.6.1 z-transformation for

difference between

two proportions

Symbol Key •  mean of population i

• mean of sampling distribution if

• sample size for sample i from population i

• proportion for population i

• proportion for sample i from population i

• variance for population i

• mean of sample i from population i

• standard normal random variablez =

Z =

Np - p

p11 - p2

x + .5

- p

2pq>n

X + .5

+ p

2pq>n

Z =

- X

2- 1m

- m

Z =

- p

2- 1p

- p

11 - p

REVIEW QUESTIONS AND EXERCISES

1. What is a sampling distribution?

2. Explain how a sampling distribution may be constructed from a ﬁnite population.

3. Describe the sampling distribution of the sample mean when sampling is with replacement from

a normally distributed population.

4. Explain the central limit theorem.

5. How does the sampling distribution of the sample mean, when sampling is without replacement,

differ from the sampling distribution obtained when sampling is with replacement?

6. Describe the sampling distribution of the difference between two sample means.

7. Describe the sampling distribution of the sample proportion when large samples are drawn.

8. Describe the sampling distribution of the difference between two sample means when large sam-

ples are drawn.

9. Explain the procedure you would follow in constructing the sampling distribution of the difference

between sample proportions based on large samples from ﬁnite populations.

10. Suppose it is known that the response time of healthy subjects to a particular stimulus is a nor-

mally distributed random variable with a mean of 15 seconds and a variance of 16. What is the

probability that a random sample of 16 subjects will have a mean response time of 12 seconds

or more?

11. Janssen et al. (A-10) studied Americans ages 60 and over. They estimated the mean body mass index

of women over age 60 with normal skeletal muscle to be 23.1 with a standard deviation of 3.7. Using

these values as the population mean and standard deviation for women over age 60 with normal skele-

tal muscle index, ﬁnd the probability that 45 randomly selected women in this age range with normal

skeletal muscle index will have a mean BMI greater than 25.

12. In the study cited in Review Exercise 11, the researchers reported the mean BMI for men ages

60 and older with normal skeletal muscle index to be 24.7 with a standard deviation of 3.3.

Using these values as the population mean and standard deviation, find the probability that 50

randomly selected men in this age range with normal skeletal muscle index will have a mean

BMI less than 24.

13. Using the information in Review Exercises 11 and 12, ﬁnd the probability that the difference in

mean BMI for 45 women and 50 men selected independently and at random from the respective

populations will exceed 3.

14. In the results published by Wright et al. (A-2) based on data from the 1999–2000 NHANES study

referred to in Exercises 5.4.1 and 5.4.2, investigators reported on their examination of iron levels. The

mean iron level for women ages 20–39 years was 13.7 mg with an estimated standard deviation of

8.9 mg. Using these as population values for women ages 20–39, ﬁnd the probability that a random

sample of 100 women will have a mean iron level less than 12 mg.

15. Refer to Review Exercise 14. The mean iron level for men between the ages of 20 and 39 years

is 17.9 mg with an estimated standard deviation of 10.9 mg. Using 17.9 and 10.9 as population

parameters, find the probability that a random sample of 120 men will have a mean iron level

higher than 19 mg.

REVIEW QUESTIONS AND EXERCISES 159

16. Using the information in Review Exercises 14 and 15, and assuming independent random samples

of size 100 and 120 for women and men, respectively, ﬁnd the probability that the difference in

sample mean iron levels is greater than 5 mg.

17. The results of the 1999 National Health Interview Survey released in 2003 (A-7) showed that

among U.S. adults ages 60 and older, 19 percent had been told by a doctor or other health care

provider that they had some form of cancer. If we use this as the percentage for all adults 65 years

old and older living in the United States, what is the probability that among 65 adults chosen at

random more than 25 percent will have been told by their doctor or some other health care provider

that they have cancer?

18. Refer to Review Exercise 17. The reported cancer rate for women subjects ages 65 and older is 17

percent. Using this estimate as the true percentage of all females ages 65 and over who have been

told by a health care provider that they have cancer, ﬁnd the probability that if 220 women are selected

at random from the population, more than 20 percent will have been told they have cancer.

19. Refer to Review Exercise 17. The cancer rate for men ages 65 and older is 23 percent. Use this

estimate as the percentage of all men ages 65 and older who have been told by a health care

provider that they have cancer. Find the probability that among 250 men selected at random that

fewer than 20 percent will have been told they have cancer.

20. Use the information in Review Exercises 18 and 19 to ﬁnd the probability that the difference in

the cancer percentages between men and women will be less than 5 percent when 220 women and

250 men aged 65 and older are selected at random.

21. How many simple random samples (without replacement) of size 5 can be selected from a popu-

lation of size 10?

22. It is estimated by the 1999–2000 NHANES (A-7) that among adults 18 years old or older 53 percent

have never smoked. Assume the proportion of U.S. adults who have never smoked to be .53.

Consider the sampling distribution of the sample proportion based on simple random samples of

size 110 drawn from this population. What is the functional form of the sampling distribution?

23. Refer to Exercise 22. Compute the mean and variance of the sampling distribution.

24. Refer to Exercise 22. What is the probability that a single simple random sample of size 110 drawn

from this population will yield a sample proportion smaller than .50?

25. In a population of subjects who died from lung cancer following exposure to asbestos, it was found

that the mean number of years elapsing between exposure and death was 25. The standard devia-

tion was 7 years. Consider the sampling distribution of sample means based on samples of size

35 drawn from this population. What will be the shape of the sampling distribution?

26. Refer to Exercise 25. What will be the mean and variance of the sampling distribution?

27. Refer to Exercise 25. What is the probability that a single simple random sample of size 35 drawn

from this population will yield a mean between 22 and 29?

28. For each of the following populations of measurements, state whether the sampling distribution of

the sample mean is normally distributed, approximately normally distributed, or not approximately

normally distributed when computed from samples of size (A)10, (B) 50, and (C) 200.

(a) The logarithm of metabolic ratios. The population is normally distributed.

(b) Resting vagal tone in healthy adults. The population is normally distributed.

160 CHAPTER 5 SOME IMPORTANT SAMPLING DISTRIBUTIONS

29. For each of the following sampling situations indicate whether the sampling distribution of the

sample proportion can be approximated by a normal distribution and explain why or why not.

(a) (b)

(e) (f)

REFERENCES

Methodology References

1. R

ICHARD

J. L

ARSEN

and M

ORRIS

L. M

ARX

, An Introduction to Mathematical Statistics and Its Applications,

2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1986.

2. J

OHN

A. R

ICE

, Mathematical Statistics and Data Analysis, 2nd ed., Duxbury, Belmont, CA, 1995.

Applications References

A-1. The Third National Health and Nutrition Examination Survey, NHANES III (1988–94), Table 2. National Cen-

ter for Health Statistics, Division of Data Services, Hyattsville, MD. Available at http://www.cdc.gov/nchs/

about/major/nhanes/datatblelink.htm.

A-2. J

ACQUELINE

D. W

RIGHT

, C

HIA

-Y

ANG

, J

OCELYN

ENNEDY

-S

TEPHENSON

, and R. B

ETHENE

RVIN

, “Dietary

Intake of Ten Key Nutrients for Public Health, United States: 1999–2000,” National Center for Health Statis-

tics. Advance Data from Vital and Health Statistics, No. 334 (2003).

A-3. Health, United States, 2002, Table 70: Healthy weight, overweight, and obesity among persons 20 years of age

and over, according to sex, age, race, and Hispanic origin: United States, 1960–62, 1971–74, 1976–80,

1988–1994, and 1999–2000. National Center for Health Statistics, Division of Data Services, Hyattsville, MD,

20782. http://www.cdc.gov/nchs/products/pubs/pubd/hestats/obese/obse99.htm.

A-4. B

LANCHE

IKHAIL

, “Prenatal Care Utilization among Low-Income African American Women,” Journal of Com-

munity Health Nursing, 17 (2000), 235–246.

A-5. J

AMES

P. S

MITH

, R

AJENDRA

H. M

EHTA

, S

UGATA

K. D

, T

HOMAS

SAI

, D

EAN

J. K

ARAVITE

, P

AMELA

USMAN

, D

AVID

RUCKMAN

, and K

A. E

AGLE

, “Effects of End-of-Month Admission on Length of Stay

and Quality of Care Among Inpatients with Myocardial Infarction,” American Journal of Medicine, 113

(2002), 288–293.

A-6. R

OBYN

ALLAGHER

, S

HARON

McK

INLEY

, and K

ATHLEEN

RACUP

, “Predictor’s of Women’s Attendance at

Cardiac Rehabilitation Programs,” Progress in Cardiovascular Nursing, 18 (2003), 121–126.

A-7. J. R. P

LEIS

and R. C

OLES

, “Summary Health Statistics for U.S. Adults: National Health Interview Survey, 1999,”

National Center for Health Statistics. Vital and Health Statistics, 10 (212), (2003).

A-8. U.S. Census Bureau, Current Population Reports, P60–215, as reported in Statistical Abstract of the United

States: 2002 (118th edition), U.S. Bureau of the Census, Washington, DC, 2002, Table Nos. 137–138.

A-9. U.S. Bureau of Labor Statistics, News, USDL 01–473, as reported in Statistical Abstract of the United States:

2002 (118th edition), U.S. Bureau of the Census, Washington, DC, 2002, Table No. 139.

A-10. I

ANSSEN

, S

TEVEN

B. H

EYMSFIELD

, and R

OBERT

OSS

, “Low Relative Skeletal Muscle Mass (Sacopenia)

in Older Persons Is Associated with Functional Impairment and Physical Disability,” Journal of the American

Geriatrics Society, 50 (2002), 889–896.

p = .05, n = 150p = .90, n = 100

p = .01, n = 1000p = .10, n = 30

p = .40, n = 30p = .50, n = 8

REFERENCES 161

162

CHAPTER OVERVIEW

This chapter covers estimation, one of the two types of statistical inference.

As discussed in earlier chapters, statistics, such as means and variances, can

be calculated from samples drawn from populations. These statistics serve as

estimates of the corresponding population parameters. We expect these esti-

mates to differ by some amount from the parameters they estimate. This

chapter introduces estimation procedures that take these differences into ac-

count, thereby providing a foundation for statistical inference procedures

discussed in the remaining chapters of the book.

TOPICS

6.1 INTRODUCTION

6.2 CONFIDENCE INTERVAL FOR A POPULATION MEAN

6.3 THE

DISTRIBUTION

6.4 CONFIDENCE INTERVAL FOR THE DIFFERENCE BETWEEN TWO POPULATION

MEANS

6.5 CONFIDENCE INTERVAL FOR A POPULATION PROPORTION

6.6 CONFIDENCE INTERVAL FOR THE DIFFERENCE BETWEEN TWO POPULATION

PROPORTIONS

6.7 DETERMINATION OF SAMPLE SIZE FOR ESTIMATING MEANS

6.8 DETERMINATION OF SAMPLE SIZE FOR ESTIMATING PROPORTIONS

6.9 CONFIDENCE INTERVAL FOR THE VARIANCE OF A NORMALLY DISTRIBUTED

POPULATION

6.10 CONFIDENCE INTERVAL FOR THE RATIO OF THE VARIANCES OF TWO

NORMALLY DISTRIBUTED POPULATIONS

6.11 SUMMARY

CHAPTER

ESTIMATION

6.1 INTRODUCTION 163

LEARNING OUTCOMES

After studying this chapter, the student will

1. understand the importance and basic principles of estimation.

2. be able to calculate interval estimates for a variety of parameters.

3. be able to interpret a conﬁdence interval from both a practical and a probabilistic

viewpoint.

4. understand the basic properties and uses of the

distribution, chi-square distri-

bution, and

distribution.

6.1 INTRODUCTION

We come now to a consideration of estimation, the ﬁrst of the two general areas of statisti-

cal inference. The second general area, hypothesis testing, is examined in the next chapter.

We learned in Chapter 1 that inferential statistics is deﬁned as follows.

DEFINITION

Statistical inference is the procedure by which we reach a conclusion

about a population on the basis of the information contained in a

sample drawn from that population.

The process of estimation entails calculating, from the data of a sample, some sta-

tistic that is offered as an approximation of the corresponding parameter of the popula-

tion from which the sample was drawn.

The rationale behind estimation in the health sciences ﬁeld rests on the assump-

tion that workers in this ﬁeld have an interest in the parameters, such as means and pro-

portions, of various populations. If this is the case, there is a good reason why one must

rely on estimating procedures to obtain information regarding these parameters. Many

populations of interest, although ﬁnite, are so large that a 100 percent examination would

be prohibitive from the standpoint of cost.

Suppose the administrator of a large hospital is interested in the mean age of patients

admitted to his hospital during a given year. He may consider it too expensive to go

through the records of all patients admitted during that particular year and, consequently,

elect to examine a sample of the records from which he can compute an estimate of the

mean age of patients admitted that year.

A physician in general practice may be interested in knowing what proportion of

a certain type of individual, treated with a particular drug, suffers undesirable side effects.

No doubt, her concept of the population consists of all those persons who ever have been

or ever will be treated with this drug. Deferring a conclusion until the entire population

has been observed could have an adverse effect on her practice.

These two examples have implied an interest in estimating, respectively, a popula-

tion mean and a population proportion. Other parameters, the estimation of which we

will cover in this chapter, are the difference between two means, the difference between

two proportions, the population variance, and the ratio of two variances.

164 CHAPTER 6 ESTIMATION

We will ﬁnd that for each of the parameters we discuss, we can compute two types

of estimate: a point estimate and an interval estimate.

DEFINITION

A point estimate is a single numerical value used to estimate the

corresponding population parameter.

DEFINITION

An interval estimate consists of two numerical values deﬁning a range

of values that, with a speciﬁed degree of conﬁdence, most likely

includes the parameter being estimated.

These concepts will be elaborated on in the succeeding sections.

Choosing an Appropriate Estimator Note that a single computed value

has been referred to as an estimate. The rule that tells us how to compute this value, or

estimate, is referred to as an estimator. Estimators are usually presented as formulas. For

example,

is an estimator of the population mean, The single numerical value that results from

evaluating this formula is called an estimate of the parameter

In many cases, a parameter may be estimated by more than one estimator. For

example, we could use the sample median to estimate the population mean. How then

do we decide which estimator to use for estimating a given parameter? The decision is

based on an objective measure or set of criteria that reﬂect some desired property of a par-

ticular estimator. When measured against these criteria, some estimators are better than oth-

ers. One of these criteria is the property of unbiasedness.

DEFINITION

An estimator, say, T, of the parameter is said to be an unbiased

estimator of if

is read, “the expected value of T.” For a ﬁnite population, is obtained

by taking the average value of T computed from all possible samples of a given size that

may be drawn from the population. That is, For an inﬁnite population,

is deﬁned in terms of calculus.

In the previous chapter we have seen that the sample mean, the sample proportion,

the difference between two sample means, and the difference between two sample

proportions are each unbiased estimates of their corresponding parameters. This prop-

erty was implied when the parameters were said to be the means of the respective sam-

pling distributions. For example, since the mean of the sampling distribution of is equalx

E1T2E1T2= m

E1T2E1T2

E(T )  U.U

to we know that is an unbiased estimator of The other criteria of good estima-

tors will not be discussed in this book. The interested reader will ﬁnd them covered in

detail in most mathematical statistics texts.

Sampled Populations and Target Populations The health researcher

who uses statistical inference procedures must be aware of the difference between two

kinds of population—the sampled population and the target population.

DEFINITION

The sampled population is the population from which one actually

draws a sample.

DEFINITION

The target population is the population about which one wishes to

make an inference.

These two populations may or may not be the same. Statistical inference procedures

allow one to make inferences about sampled populations (provided proper sampling meth-

ods have been employed). Only when the target population and the sampled population

are the same is it possible for one to use statistical inference procedures to reach conclu-

sions about the target population. If the sampled population and the target population are

different, the researcher can reach conclusions about the target population only on the

basis of nonstatistical considerations.

Suppose, for example, that a researcher wishes to assess the effectiveness of some

method for treating rheumatoid arthritis. The target population consists of all patients

suffering from the disease. It is not practical to draw a sample from this population. The

researcher may, however, select a sample from all rheumatoid arthritis patients seen in

some speciﬁc clinic. These patients constitute the sampled population, and, if proper sam-

pling methods are used, inferences about this sampled population may be drawn on the

basis of the information in the sample. If the researcher wishes to make inferences about

all rheumatoid arthritis sufferers, he or she must rely on nonstatistical means to do so.

Perhaps the researcher knows that the sampled population is similar, with respect to all

important characteristics, to the target population. That is, the researcher may know that

the age, sex, severity of illness, duration of illness, and so on are similar in both popu-

lations. And on the strength of this knowledge, the researcher may be willing to extrap-

olate his or her ﬁndings to the target population.

In many situations the sampled population and the target population are identical;

when this is the case, inferences about the target population are straightforward. The

researcher, however, should be aware that this is not always the case and not fall into

the trap of drawing unwarranted inferences about a population that is different from the

one that is sampled.

Random and Nonrandom Samples In the examples and exercises of this

book, we assume that the data available for analysis have come from random samples.

The strict validity of the statistical procedures discussed depends on this assumption. In

m.x

6.1 INTRODUCTION 165

166 CHAPTER 6 ESTIMATION

many instances in real-world applications it is impossible or impractical to use truly ran-

dom samples. In animal experiments, for example, researchers usually use whatever ani-

mals are available from suppliers or their own breeding stock. If the researchers had to

depend on randomly selected material, very little research of this type would be conducted.

Again, nonstatistical considerations must play a part in the generalization process.

Researchers may contend that the samples actually used are equivalent to simple random

samples, since there is no reason to believe that the material actually used is not represen-

tative of the population about which inferences are desired.

In many health research projects, samples of convenience, rather than random sam-

ples, are employed. Researchers may have to rely on volunteer subjects or on readily avail-

able subjects such as students in their classes. Samples obtained from such sources are exam-

ples of convenience samples. Again, generalizations must be made on the basis of

nonstatistical considerations. The consequences of such generalizations, however, may be

useful or they may range from misleading to disastrous.

In some situations it is possible to introduce randomization into an experiment even

though available subjects are not randomly selected from some well-deﬁned population. In

comparing two treatments, for example, each subject may be randomly assigned to one or

the other of the treatments. Inferences in such cases apply to the treatments and not the sub-

jects, and hence the inferences are valid.

6.2 CONFIDENCE INTERVAL

FOR A POPULATION MEAN

Suppose researchers wish to estimate the mean of some normally distributed population.

They draw a random sample of size n from the population and compute which they use

as a point estimate of Although this estimator of possesses all the qualities of a

good estimator, we know that because random sampling inherently involves chance,

cannot be expected to be equal to

It would be much more meaningful, therefore, to estimate by an interval that

somehow communicates information regarding the probable magnitude of

Sampling Distributions and Estimation To obtain an interval estimate,

we must draw on our knowledge of sampling distributions. In the present case, because we

are concerned with the sample mean as an estimator of a population mean, we must recall

what we know about the sampling distribution of the sample mean.

In the previous chapter we learned that if sampling is from a normally distributed

population, the sampling distribution of the sample mean will be normally distributed with

a mean equal to the population mean and a variance equal to We could

plot the sampling distribution if we only knew where to locate it on the -axis. From our

knowledge of normal distributions, in general, we know even more about the distribution

of in this case. We know, for example, that regardless of where the distribution of is

located, approximately 95 percent of the possible values of constituting the distribution

are within two standard deviations of the mean. The two points that are two standard devi-

ations from the mean are and so that the interval will con-

tain approximately 95 percent of the possible values of We know that and, hence

are unknown, but we may arbitrarily place the sampling distribution of on the -axis.x

,mx.

m ; 2s

m + 2s

,m - 2s

>n.s

m,m

mm.