Estimation in Statistics
In statistics, estimation refers to the process by
which one makes inferences about a population, based on information
obtained from a sample.
Point Estimate vs. Interval Estimate
Statisticians use sample
statistics
to estimate population
parameters.
For example,
sample means are used to estimate population means; sample proportions,
to estimate population proportions.
An estimate of a population parameter may be expressed in two ways:
- Point estimate. A point estimate of a population
parameter is a single value of a statistic. For example,
the sample mean x
is a point estimate of the population mean μ. Similarly,
the sample proportion p is a point estimate of the
population proportion P.
- Interval estimate. An interval estimate is
defined by two numbers, between which a population parameter
is said to lie. For example, a
< x < b is an
interval estimate of the
population mean μ. It indicates that the population
mean is greater than a but less than b.
Confidence Intervals
Statisticians use a confidence interval to express the
precision and uncertainty associated with a particular
sampling method.
A confidence interval consists of three parts.
- A confidence level.
- A statistic.
- A margin of error.
The confidence level describes the uncertainty of a sampling method.
The statistic and the margin of error define an interval estimate
that describes the precision of the method. The interval estimate
of a confidence interval is defined by the
sample statistic + margin of error.
For example, suppose we compute an interval estimate of a population
parameter. We might describe this interval estimate as a 95%
confidence interval. This means that if we used the same
sampling method to select different samples and compute different
interval estimates, the true
population parameter would fall within a range defined by the
sample statistic + margin of error
95% of the time.
Confidence intervals are preferred to point estimates,
because confidence intervals indicate (a) the
precision of the estimate and (b) the uncertainty of the estimate.
Confidence Level
The probability part of a confidence interval is called a
confidence level. The confidence level describes
the likelihood that a particular sampling method will
produce a confidence interval that includes the true population
parameter.
Here is how to interpret a confidence level.
Suppose we collected all possible samples from a given population,
and computed confidence intervals for each
sample. Some confidence intervals would include the true
population parameter; others would not. A 95% confidence level
means that 95% of the intervals contain the
true population parameter; a 90% confidence level means that
90% of the intervals contain the population parameter; and so on.
Margin of Error
In a confidence interval, the range of values above and below the
sample statistic is called the
margin of error.
For example, suppose the local newspaper conducts an election survey and
reports that the independent candidate will receive 30% of
the vote. The newspaper states that the survey had a 5% margin of
error and a confidence level of 95%. These findings result in the
following confidence interval: We are 95% confident that the
independent candidate will receive between 25% and 35% of the vote.
Note: Many public opinion surveys report interval
estimates, but not confidence intervals. They provide the margin
of error, but not the confidence level. To clearly interpret survey
results you need to know both! We are much more likely to accept
survey findings if the confidence level is high (say, 95%)
than if it is low (say, 50%).
Test Your Understanding
Problem 1
Which of the following statements is true.
I. When the margin of error is small, the confidence level is high.
II. When the margin of error is small, the confidence level is low.
III. A confidence interval is a type of point estimate.
IV. A population mean is an example of a point estimate.
(A) I only
(B) II only
(C) III only
(D) IV only.
(E) None of the above.
Solution
The correct answer is (E). The confidence level is not affected
by the margin of error. When the margin of error is small, the
confidence level can low or high or anything in between. A
confidence interval is a type of interval estimate, not a type of point
estimate. A population mean is not an example of a point
estimate; a sample mean is an example of a point estimate.