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 of a population parameter 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 plan. A confidence interval consists of three parts: a statistic, a margin of error, and a confidence level.

• A statistic. The statistic is a point estimate of a parameter (e.g., a sample mean or a sample proportion)
• A margin of error. The margin of error describes a range of values around the statistic.
• A confidence level. The confidence level describes the probability that a sampling plan will generate an interval estimate that includes the true population parameter.

The interval estimate of a confidence interval is defined by the sample statistic ± margin of error.

How to Interpret a Confidence Level

Here is how to interpret a confidence level. Suppose we collected all possible random samples of size n from a given population, and computed a confidence interval 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 would contain the true population parameter; a 90% confidence level means that 90% of the intervals would contain the population parameter; and so on.

A confidence level describes the probability that a sampling plan that will produce a confidence interval that contains the true popuation parameter. The confidence level does not describe the probability that the true population parameter falls within a particular interval estimate.

For example, suppose the local newspaper conducts a random sample of voters 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%. Or the newspaper might say: We are 95% confident that the independent candidate will receive between 25% and 35% of the vote.

This does not mean there is a 95% chance that the true level of support for the independent candidate ranges between 25% and 35% of the vote. The true level of support is a constant. The probability that a constant ranges between 25% and 35% is either zero or one. A 95% confidence level refers to the quality of the sampling plan. It means the sampling plan used by the newspaper should produce a confidence interval that includes the true level of voter support 95% of the time.

Note: Many public opinion surveys report an interval estimate (a sample statistic and the margin of error), but not the confidence level. To clearly interpret survey results you need to know the sample statistic, the margin of error, and the confidence level! We are much more likely to accept survey findings if the confidence level is high (say, 95%) than if it is low (say, 75%).

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 be 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.