Inferential statistics-estimation

Lecture Series on
Biostatistics

Inferential Statistics-
Estimation
By
Dr. Bijaya Bhusan Nanda,
M. Sc (Gold Medalist) Ph. D. (Stat.)
Topper Orissa Statistics & Economics Services, 1988
bijayabnanda@yahoo.com

CONTENTS
 Introduction
 Confidential interval for a population mean
 The t distribution
 Confidence interval for the difference
between two population mean
 Confidence interval for a population
proportion
 Confidence interval for the difference
between two population proportion

Introduction
Statistical Inference
 It is the procedure by which we reach a conclusion
about a population on the basis of information
contained in the sample drawn from that population.
 Two broad areas of statistical inference
 Estimation
 Hypothesis testing.
 Estimation- It is the process of calculating some
statistics based upon the sample data drawn from a
certain population. The statistics is used as an
approximation to the population parameter.

Types of Estimate
For each of parameters, we can have
 Point Estimate
 Interval estimate
 A point estimate is a single numerical value used to
estimate the corresponding population parameter.
An interval estimate consists of two numerical values
defining a range of values that, with a specified degree
of confidence, includes the parameter being estimated.

Choosing an Appropriate Estimation
 A single computed value is an estimate.
 An estimator usually presented as a formula. For
example
x=
∑ xi
n is the estimator of population mean
 The single numerical value that results from
evaluating this formula is called an estimate of the
parameter µ.
 E(T) is obtained by taking the average value of T
computed from all possible sample i.e. E(T)= µT

Criteria of a good estimator:
There can be more than one estimator for the parameter.
For example population mean can be estimated by the
sample median or sample mean.
But a good estimator should fulfill certain criteria. One such
criterion of a good estimator is unbiasedness.
An estimator T of the parameter θ is said to be an unbiased
estimator of θ if E(T)= θ
 Sample mean is an unbiased estimator of population mean.
 Sample proportion is an unbiased estimate of population
proportion.
 The difference between two sample means is an unbiased
estimate of difference between the population mean.
The difference between two sample proportions is an
unbiased estimate of difference between the population
proportion.

Sampled Population and Target Population
 Sampled population is the population from which one
actually draws a sample.
 The target population is the population about which
one wishes to make inference.
 Statistical inference procedure allows one to make
inference about sampled population (provided proper
sampling methods have been employed)
 Only when sampled population and the target
population are the same, it is possible to reach
statistical inference about the target population.
Random and Non random Samples
 The strict validity of the statistical procedures
discussed depends on the assumptions of random
sample.

 In real world applications it is impossible or
impractical to use truly random samples.
 If the researchers had to depend on randomly selected
materials, very little research of this type would be
conducted.
 Therefore, non statistical considerations must play a
past in the generalizations process.
 Researchers may contend that samples actually used
are equivalent to simple random samples, since there is
no reasons to believe that material actually used is not
representative of the population about which inferences
are desired.
 In many health research projects, samples of
convenience, rather than random samples are
employed.

 Generalization must be made on non statistical
consideration.
 Consequences of such generalization, however may
range from misleading to disastrous.
 In some situation it is possible to introduce
randomization into experiment even though available
subjects are not randomly selected from well defined
population.
 Example: Allocating subjects to treatments in a random
manner.

Confidential interval for a population mean
 Say x the sample mean from a random sample of size n
is
drawn from a normally distributed population.
 x is an unbiased estimator of the population mean
 ()
E x =µ
 But because of sampling fluctuation x can’t be expected
to be equal to µ.
 Therefore, we should have an interval estimate µ.
 For the interval estimate we must have knowledge of
sampling distribution of x .
 Sampling distribution of x for a normal population is as
follows

x ≈ N ( µ , σ )i.e.N ( µ ,σ )
n
x x

 Following the normal probability distribution the
interval estimate for µ is as follows
95% confidence interval for µ = µ ± 2σ x
since we don’t know µ with the help of sample mean x we
have the 95% confidence interval for µ as
x ± 2σx = x ± 2 σ n
If we don’t know σ the 95% confidence interval for
µ = x ± 2 s / n where s= the standard deviation based
on the sample.

Interval estimate Component
 In general an interval estimate may be expressed as
follows
estimator ± (reliability coefficient)× (standard error)
 In particular when sampling is from a normal
distribution with known variance, an interval
estimator for µ may be expressed as
x±z σ
(1 − α / 2 ) x
where z(1-α /2) is the value of z to the left of which lies
1- α /2 and to the right of which lies α /2 of the area
under its curve.

Interpreting Confidence Interval:
 Probabilistic Interpretation: In repeated sampling,
from a normally distributed population with a known
standard deviation, 100(1- α ) percent of all intervals of
the form x±z σ
(1 − α / 2 ) x will in the long run
include the population mean µ .
 Practical Interpretation: When sampling is from a
normally distributed population with known standard
deviation, we are 100(1- α ) % confident that the single
computed interval,
population mean µ .
x±z σ
(1 − α / 2 ) x
, contains the

 Precision: The quantity obtained by multiplying the
reliability factor by the standard error of the mean is
called the precision of the estimate. This quantity is also
called the margin of error.

Example
A physical therapist wished to estimate, with 99 percent
confidence , the mean maximum strength of a particular
muscle in a certain group of individuals. He is willing to
assume that strength scores are approximately normally
distributed with a variance of 144. A sample of 15 subjects
who participated in the experiment yielded a mean of 84.3.
Solution:
The z value corresponding to a confidence coefficient of 0.99
is found in Table D to be 2.58. This is our reliability
coefficient. The standard error is σx = 12 = 3.0984
.Our 99% 15
84.3 ± 2.58(3.0984)
Confidence interval for µ, then
84.3 ± 8.0
76.3,92.3

We say we are 99% confidence that the
population mean is between 76.3 and 92.3
since, in repeated sampling, 99% of all
intervals that could be constructed in the
manner just described would include the
population mean.

The t distribution
 Estimation of confidence interval using standard normal z
distribution applies when population variance is known.
Usually, the knowledge of population variance is not
known.
 This condition presents a problem to construct confidence
intervals.
x −µ
Although the statistic z = σ is normally distributed
n
or approximately normally distributed when n is large,
regardless of the functional form of the population, we
can’t make use of this fact because σ is not known.

 In this case we may use of
s= ∑ ( xi − x )
2
( n − 1)
as an estimator of σ. When sample size is large, say,
greater than 30, s is a good approximate of σ is
substantial and we may use normal distribution theory
to construct confidence interval.
 When the sample size is small, ≤30, we make use of
Student’s t distribution.

x−µ
t=
The quantity s follows this distribution.
n

Properties of the t distribution:
 It has a mean of 0.
 Symmetrical about the mean.
 In general, it has a variance greater than 1, but the
variance approaches 1 as the sample size becomes
large. For df >2, the variance for t distribution is
df /(df-2).
 The variable t ranges from -∞ to + ∞ .
 The t distribution is really a family of distributions,
since there is a different distribution for each sample
value of n-1, the divisor used in computing s2. We recall
that n-1 is referred to as degrees of freedom.
 Compared to normal distribution is less peaked at
center and has higher tails t distribution approaches
the normal distribution an n-1 approaches infinity.
 The t distribution approaches the normal distribution
as n-1 approaches infinity.

The t distribution for different degrees of freedom
Degrees of freedom=30

Comparison of normal distribution and t distribution

Normal Distribution

t distribution

Confidence Intervals Using t:
estimator ± (reliability coefficient) (standard error)
error
The source of reliability co-efficient is ‘t’ distribution
rather than standard normal z distribution.
 When sampling is from a normal distribution whose
standard deviation , σ , is unknown the 100(1-α )%
confidence interval for the population mean µ , is
given by
s
x ± t (1 − α / 2 )
n
 A moderate departures of the sampled population
from the normally can be tolerated in using the ‘t’
distribution. An assumption of mound shaped
population distribution be tenable.

Example:
Maureen McCauley conducted a study to evaluate the effect
of on the job body mechanics instruction on the work
performance of newly employed young workers (A-1). She
used two randomly selected groups of subjects, an
experimental group and a control group. The experimental
group received one hour of back school training provided on
occupational therapist. The control group did not receive this
training. A criterion referenced Body Mechanics Evaluation
Checklist was used to evaluate each workers lifting, lowering,
pulling, and transferring of objects in the work environment.
A correctly performed task received a score of 1. The 15
control subjects made a mean score of 11.53 on the evaluation
with a standard deviation of 3.681. We assume that these 15
controls behave as a random sample from a population of
similar subjects. We wish to use these sample data to estimate
the mean score fro the population.

Solution:
We may assume sample mean, 11.53, as a point estimate
of population mean but since the population standard
deviation is unknown, we must assume the population of
values to be at least approximately normally distributed
before constructing a confidence interval for µ .
Let us assume an assumption is reasonable and that a
95% confidence interval is desired. Our estimator x is
and our standard error is

s n = 3.681 = 0.9564
15
We need to find reliability coefficient, the value of t
associated with a confidence coefficient of 0.95 and n-
1=14 degrees of freedom.

Confidence interval is equally divided in to two tails
i.e.0.025 .In Table E the tabulated value of t is 2.1448, which
is our reliability coefficient.
Then our 95% confidence interval is as follows
11.53 ± 2.1448(0.9504)
11.53 ± 2.04
9.49,13.57
Deciding Between z and t:
To make an appropriate choice between z and
we must consider whether the sampled population is normally
distributed, and whether the population variance is known.

Flowchart for use in deciding between z and t when
making inferences about population means
Normal
Y Population N

Large Large
Y Sample N Y Sample N

Populn Y Populn N Y Populn N Y Populn N
Y N
var var var var
known?
known? known? known?

z t z t z z * *

z Central limit theorem applies

* = use nonparametric Procedure

Confidence interval for the difference between two
population means
Normal Population
 Say µ -µ = the difference between two normal population
1 2
means.

x −x
1 2 = the difference between two independent
sample means drawn from the two populations respectively.
E ( x 1 − x 2) = µ1 − µ 2
σ1 σ 2
2
V ( x 1 − x 2) = + 2
n1 n 2
Where σ12 & σ22 are the variance of the population 1&2 and n1,
n2 are the size of the SRS drawn from population 1&2
respectively.

 When the population variances are known, the 100(1-
α)% confidence interval for µ1& µ2 is given by

σ σ
2 2

(x − x ) ± z +
1 2
1 2 (1 − α / 2 )

n 1 n 2

 When the confidence interval includes zero, we say
that the population means may be equal when it
doesn’t includes zero we say that the interval
provides evidence that the two population means are
not equal.

Example:
A research team is interested in the difference between
serum uric acid levels in patients with and without Down’s
syndrome. In a large hospital for the treatment of the
mentally retarded, a sample of 12 individuals with Down’s
syndrome yielded a mean of 4.5mg/100ml. In a general
hospital a sample of 15 normal individuals of the same age
and sex were found to have a mean value of 3.4. If it is
reasonable to assume that the two populations of values are
normally distributed with variances equal to 1 and 1.5, find
the 95% confidence interval for µ1- µ2.
Solution:
For a point estimate of µ1- µ2, we use
x 1 − x 2 = 4.5 − 3.4 = 1.1

The reliability coefficient corresponding to 0.95 is found in
Table D to be 1.96. The standard error
σ
2
σ 1 1 .5
2
σ ( x 1 − x 2) = +1
= +2
= 0.4282
n1 n2 12 15
The 95 percent confidence interval, then, is
1.1 ±1.96(0.4282)
1.1 ± 0.84
0.26,1.94
The difference µ1- µ2, is some where between 0.26 and 1.94,
because, in repeated sampling, 95% of the intervals
constructed in this manner would include the difference
between the true means. Since the interval does not include
zero, we conclude that the two population means are not
equal.

Sampling from Nonnormal Population:
If sample sizes n1 and n2 are large, we may construct the
confidence interval the same way as we do incase of
normal population in accordance with the Central Limit
Theorem.
Example:
Motivated by an awareness of the existence of a body of
controversial literature suggesting that stress, anxiety, and
depression are harmful to the immune system, Gorman et al.
(A-5) conducted a study in which the subjects were homosexual
men and some of whom were HIV positive and some of whom
were HIV negative. Data were collected on a wide variety of
medical, immunological, psychiatric, and neurological measures,
one of which was the number of CD4+ cells in the blood. The
mean number of CD4+ cells for the 112 men with HIV infection
was 401.8 with a standard deviation of 226.4. For the 75 men
without HIV infection the mean and standard deviation were
828.2 and 274.9, respectively. We wish to construct a 99%
confidence interval for the difference between population
means.

Solution:
Here we are using z statistic as the reliability factor in the
construction of our confidence interval. Since the population
standard deviation are not given, we will use the sample standard
deviations to estimate them. The point estimate for the difference
between population means is the difference between sample
means, 882.2-401.8=426.4. In Table D, we find the reliability
factor to be 2.58. The estimated standard error is

2 2
274.9 226.4
sx − x = + = 38.279
75 112
1 2

Our 99% confidence interval for the difference between
population means is
426.4 ± 2.58(38.279)
327.6,525.2
We are 99% confident that the mean number of CD4+
cells in HIV +ve males differs from the mean for HIV
–ve males by some where between 327.6 and 525.2.

Confidence interval using ‘t’ distribution:
 Say the variances of the two sampled population are not
known.
 The two sampled population are normally distributed.
 The difference between the two population means can be
estimated through confidence interval using ‘t’ distribution
as the source of reliability factor.
Situation-I-Equal population variances:
The 100(1-α)% confidence interval for µ1- µ2 is given by
2 2
s s
( x 1 − x 2) ± t (1 − α / 2 ) p
+ p

n1 n2
If the two sample sizes are unequal , the weighted average
takes advantages of the additional information provided by
the larger sample. The pooled estimate is given by

( n1 −1) s + ( n 2 −1) s
2 2

s
2
p = 1 2

n1 + n 2 − 2
The standard error of the estimate , then, is given by

2 2
s s
Sx 1 − x2 = p
+ p

n1 n2
The ‘t’ follows a Student’s t distribution with n1+n2-2 d.f.
Example:
The purpose of a study by Stone et al.(A-6) was to
determine the effects of long-term exercise intervention on
corporate executives enrolled in a supervised fitness program.

Data were collected on 13 subjects (the exercise group) who
voluntarily entered a supervised exercise program and remained
active for an average of 13 years and 17 subjects (the secondary
group) who elected not to join the fitness program. Among
the data collected on the subjects was maximum number of
sit-ups completed in 30 seconds. The exercise group had a
mean and standard deviation for this variable of 21.0 and
4.9 respectively. The mean and standard deviation for the
secondary group were 12.1 and 5.6 respectively. We assume
that the two population of overall muscle condition
measures are approximately normally distributed and that
the two population variances are equal. We wish to
construct a 95% confidence interval for the difference
between the means of the populations represented by these
two samples.
Solution:
The 95% confident that the difference between
population means is somewhere between 4.9, 12.9 .

Situation-II-Unequal population variances:
An approximate 100(1-α)% confidence interval for µ1- µ2 is
given by 2
s1 s 2
2
x 1 − x 2 ± t ′(1 − α / 2 ) +
n1 n2
The solution proposed by Cochran consists of computing the
reliability factor t ′ =
w1t1 + w2 t 2
(1−α 2)
w1 + w2
Where w1=s12/n1, w2=s22/n2, t1=t1-α/2 for n1-1 degrees of freedom,
and t2=t1- α/2 for n2-1 degrees of freedom.
Example:
in the study by Stone et al.(A-6)described in previous
example , the investigator also reported the following
information on a measure of overall muscle condition
scores made by the subjects:

Sample n Mean S.D
Exercise 13 4.5 0.3
group
Sedentary 17 3.7 1.0
group

We assume that the two populations of overall muscle
condition scores are approximately normally distributed. We
are unwilling to assume, however, that the two population
variances are equal. We wish to construct a 95% confidence
interval for the difference between the mean overall muscle
condition scores of the two populations represented by the
samples.
Solution:
The 95% confidence interval for the difference between the
two population means when they are not equal is 0.25, 1.34

Flowchart in deciding whether the reliability factor should be z, t, or t’
when making inferences about the difference between two population
means Normal
Y Population N

Y
Large N Large
Sample Y N
Sample

Populn Populn Populn Populn
Y N Y N Y N Y N
var var var var
known? known? known? known?

=? =? =? =? =? =? =? =?

Y N Y N Y N Y N Y N Y N Y N Y N

z z t t’ z z t t’ z z z t’ * * * *
or or * = Nonparametric test to be applied
z z

Confidence interval for a population proportion
 Population proportion in many situations is a matter of
great interest to the health professional.
 What proportion of people suffers from different diseases,
disability etc.? what proportion of patience response to a
particular treatment? These are certain important questions
for the health purpose.
 We estimate the population proportion as in the case of
population mean.
 A random sample is drawn from the population of interest
and the sample proportion p provides an unbiased estimate
ˆ
of the population proportion P.
 A confidence interval is obtained by the general formula
estimator ± (reliability coefficient) × (standard error)
 when both np and n(1-p) are greater than 5, the sampling
distribution of p is quite close to the normal distribution.
ˆ

 When above condition is mate100(1-α) % confidence
interval for p is given by

p ± z (1 − α / 2 )
ˆ p(1 − p )
ˆ ˆ
Example:
n
Mathers et al.(A-12) found that in a sample of 591 patients
admitted to a psychiatric hospital, 204 admitted to using
cannabis at least once in their lifetime. We wish to
construct a 95% confidence interval for the proportion of
lifetime cannabis users in the sampled population of
psychiatric hospital admission.
Solution:
The 95% confident that the population proportion p is
between 0.3069, 0.3835

Confidence interval for the difference between two
population proportion
 An unbiased point estimator of the difference between two
population proportions (p1-p2) is provided by the difference
ˆ ˆ
between sample proportion p1 − p 2 .
 When n1 and n2(sample size) are large and the population
proportions are not too close to 0 and 1, normal distribution
theory may be applied to obtain confidence interval.
 A 100(1-α)% confidence interval for p1-p2 is given by

p1 (1 − p1)
ˆ ˆ p 2 (1 − p 2 )
ˆ ˆ
p 2 − p 2 ± z (1 − α / 2 )
ˆ ˆ +
n1 n2

Example:
Borst et al. (A-16) investigated the relation of ego
development, age, gender and diagnosis to suicidality among
adolescent psychiatric inpatients. There sample consisted of 96
boys and 123 girls between the age of 12 and 16 selected from
admissions to a child and adolescent unit of a private
psychiatric hospital. Suicide attempts were reported by 18 of
the boys and 60 of the girls. Let us assume the girls and the
boys behave like a simple random sample from a population
of similar girls and that the boys likewise may be considered a
SRS from a population of similar boys. For these two
population, we wish to construct a 99% confidence interval for
the difference between between the proportions of suicide
attempters.
Solution:
The 99% confident that for the sampled populations , the
proportion of suicide attempts among girls exceeds the
proportion of suicide attempts among boys by somewhere
between 0.1450, 0.4556

Example:
Goldberg et al. (A-24) conducted a study to determine if an
acute dose of dextroamphetamine might have positive
effects on affect and cognition in schizophrenic patients
maintained on a regimen of haloperidol. Among the
variables measured was the change in patients’ tension-
anxiety states. For n2=4 patients who responded to
amphetamine, the standard deviation was 5.8. Let us assume
that these patients constitute independent SRS from
populations of similar patients. Let us also assume that
change scores in tension-anxiety state is a normally
distributed variable in both populations. We wish to
construct a 95% confidence interval for the ratio of the
variances of these two populations.
Solution: σ1 2

0.2081< σ2 <14.0554
2

Inferential statistics-estimation

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