Estimation

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 Estimation
• In general terms, estimation uses a sample
statistic as the basis for estimating the value of
the corresponding population parameter.
• Although estimation and hypothesis testing are
similar in many respects, they are
complementary inferential processes.
• A hypothesis test is used to determine whether
or not a treatment has an effect, while estimation
is used to determine how much effect.

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 Estimation (cont.)
• This complementary nature is
demonstrated when estimation is used
after a hypothesis test that resulted in
rejecting the null hypothesis.
• In this situation, the hypothesis test has
established that a treatment effect exists
and the next logical step is to determine
how much effect.

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 Estimation (cont.)
• It is also common to use estimation in
situations where a researcher simply
wants to learn about an unknown
population.
• In this case, a sample is selected from the
population and the sample data are then
used to estimate the population
parameters.

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 Estimation (cont.)
• You should keep in mind that even though
estimation and hypothesis testing are inferential
procedures, these two techniques differ in terms
of the type of question they address.
• A hypothesis test, for example, addresses the
somewhat academic question concerning the
existence of a treatment effect.
• Estimation, on the other hand, is directed toward
the more practical question of how much effect.

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 Estimation (cont.)
• The estimation process can produce either
a point estimate or an interval estimate.
• A point estimate is a single value and has
the advantage of being very precise.
• For example, based on sample data, you
might estimate that the mean age for
students at the state college is μ = 21.5
years.

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 Estimation (cont.)
• An interval estimate consists of a range of
values and has the advantage of providing
greater confidence than a point estimate.
• For example, you might estimate that the mean
age for students is somewhere between 20 and
23 years.
• Note that the interval estimate is less precise,
but gives more confidence.
• For this reason, interval estimates are usually
called confidence intervals.

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 The Need for Estimates
• Consider this situation:
– Suppose a researcher wishes to evaluate the
relationship between toxic reactions to drugs and
fractures from falls among the elderly
– For logistic and economic reasons, entire
population cannot be studied to determine
proportion who have toxic reactions and fractures
– Instead a cohort study could be conducted with a
random sample of elderly patients followed for a
specified period
• Proportion of patients in the sample who
experience drug reactions and fractures
can be determined and used as an
estimate of the proportion of drug
reactions and fractures in the population
• i.e. the sample proportion is an estimate of
the population proportion π
• Both sample proportion and sample mean
are called point estimates
• Because they involve a specific number
rather than an interval or a range
• Other point estimates are the sample
standard deviation SD as an estimate of σ
• Also sample correlation r is an estimate of the
population correlation ρ
 Confidence Intervals and confidence
limits
• Instead of giving a simple point estimate,
investigator may wish to indicate the variability
the estimate would have in other samples
• To indicate this variability, interval estimates
are used
• A disadvantage of point estimates such as
mean weight loss of 20kg is that they do not
have an associated probability indicating how
likely the value is
• In contrast probability can be associated
with interval estimates such as the interval
from say 15 to 25kg
• Interval estimates are called confidence
intervals
• They define upper limit (25kg) and a lower
limit (15kg) with an associated probability.
• The ends of the confidence interval (15 and
25kg) are called confidence limits
• CI can be established for any population
parameter
• Such as CI for the mean, proportion,
relative risk, odds ratio, and correlation as
well as difference between two means,
two proportions, etc.
 Confidence intervals
• Intervals containing the true estimate of a
population parameter with a stated degree of
confidence
• uses the concept of sampling distributions to
compute intervals at appropriate probability
levels
• General formula
= estimate ± (confidence coefficient) x measure
of variability of the estimate
Confidence coefficient is a number related to
the level of confidence we want
Typical values: 95%, 99%
• For 95% confidence, we want the value that
separates the central of the distribution from
the 5% in two tails
• 95% CI for the mean is:
• sample mean ± t SE
 Confidence intervals
• General formula
= estimate ± Z1-α/2 (standard error)
• 95% confidence intervals are commonly
reported
WHAT IS A CONFIDENCE INTERVAL
– Confidence interval is a range of likely values for an
unknown population parameter at a given confidence
level.
– Provides uncertainty attached to sample estimate on
account of sampling error.
– Only by convention that the 95% confidence level is
commonly chosen.
• CI: Statistic + 1.96 S.E. (Statistic).
• E.g. 95% of the distribution of sample means lies
within 1.96 SD of the population mean
 Interpretation
• If experiment is repeated many times, the
interval would contain the true population
mean on 95% of occasions

• i.e. a range of values within which we are 95%

confident that the true population mean lies
 Issues in CI interpretation
• How wide is it? A wide CI indicates that
estimate is imprecise
• A narrow one indicates a precise estimate
• Width is dependent on size of SE, which in
turn depends on SS
Advantages of confidence intervals
• Width of confidence interval (CI) depend on
standard error of the estimate and degree of
confidence.
• Gives confidence interval ability to detect lack of
precision of estimates.
• Note precision is the standard error of the sample
statistic.
 Calculating confidence limits for
Proportion

• A total of 129 women participated in an

anticoagulant study. If 89 women survived a
heart attack, what is the confidence interval for
the proportion of survival in this study?
– Note: use 95% confidence level

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 ANSWER

• P= 89/129 = 0.69
• CI: Statistic ± (Z1-α/2) × sqrt[p(1-p)/n]
Statistic ±1.96×S.E

0.69 0.31
• Standard error = 129
0.04072

• 95% confidence limits= 0.69±1.96 ×0.0407

=(0.61, 0.77)
or (61%, 77%)
• 99% confidence limits =0.69±2.58 ×0.0407
=(0.58, 0.80) 23
 Interpretation
• In other samples, if 95% CI is calculated for
each mean, 95% of these CIs would contain
the true mean.
• They can therefore have 95% confidence that
the interval calculated contains the actual
value
 QUESTION
• A total of 120 Yoruba children were screened
for myopia in a rural village and 20 of these
children were determined to be myopic.
Calculate the 95% confidence interval for the
true prevalence of myopia in this population.
 Solution
• Note SE of p =√p(1-p)/n
 Summary
• Confidence intervals are used to determine
the confidence with which we can assume
future estimates (such as the mean) will vary
in future studies.
• The z distribution is used to form confidence
intervals
• The width of CIs depends on the confidence
value. 99% CI are wider than 95% CI because
99% CI provide greater confidence.
 Factors Influencing the Width of a
Confidence Interval
• If other factors are held constant,
increasing the level of confidence (for
example, from 80% to 90%), will cause in
increase in the width of the confidence
interval.
• To obtain greater confidence, you must
use a wider range of t statistic values,
which results in a wider interval.

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 Factors Influencing the Width of a
Confidence Interval (cont.)
• On the other hand, increasing the sample
size will cause a decrease in the width of
the confidence interval.
• In simple terms, a larger sample gives
you more information, which means that
you can estimate the population parameter
with more precision.

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