Two Methods of Estimating  True Average of Populations

There are two methods of estimating the true average of a population:

1. Use a“Point Estimate”. A point estimate is a single value to represent the truth­ (or the population parameter in question). The sample mean, x , is a point estimate of the true average of a population ([latex]\mu[/latex]).
2. Use an “Interval Estimate”.  An interval estimate is an “interval” or range of values that states that the population parameter falls between this value and another value and also defines the likelihood of the parameter falling between these values with a level of certainty or probability.

Point EStimates (example)

Point estimates are widely used but they have certain flaws. For example, suppose we wanted to determine the true average starting salary of a group of students after graduation:

• One way of estimating the average salary would be to use a single value or point.
• But is there one appropriate point (or single value) to represent the true average salary of all students graduating?
• The mean of a sample could be a logical indicator of the population average but has drawbacks.

Example 44.1

Problem Setup: Let us continue with the example of estimating starting salaries for diploma students after graduation. Suppose three recent graduates’ salaries were collected and the sample mean was $68,000.

Question: Would this be the true average salary of all students who graduated?

Solution: Most likely not – it would depend which three people were interviewed!  If we took another sample of three students the sample mean would most likely be quite different!

ProblemS With Point Estimates

Some problems with point estimates are:

• We really have no idea how far off our estimate is from the true average salary!
• There are problems with using a single value or point estimate as an indication of the population average.
• They are still used today quite liberally in business and our daily lives.
• See the Harvard Business Review article on the “Flaw of Averages.”

Interval Estimates (Example)

A second way to estimate the true average of a population is to use an interval estimate:

• We will examine interval estimates in details in the next few chapters.
• They give a range of values for the mean to be between.
• It is a range that is, hopefully, narrow enough to give a useful range of values
• While being wide enough to have an acceptable certainty that the true average is contained in this interval.

Example 44.2

Problem Setup: Suppose your department hired your Statistics Instructor to determine an interval estimate of the true starting salary of all recent graduates.  After the study is completed, the instructor states the findings: “the true average salary is between negative infinity and positive infinity”.

Question: Can this statement be true?

Solution: Ha! Well, let’s think about this for a minute.

• Can this statement be true? Possibly yes.
• Is this statement useful? Definitely not!

Conclusion: What we hope to achieve with an interval estimate is:

• To come up with a trade-off between the two extremes of a point estimate (that is most likely to be incorrect)
• And an interval estimate (that is so wide that it has no practical usefulness).
• Ie: Come up with a statement that is narrow enough to be useful AND has enough certainty that we can live with it.

So, although the instructor’s statement is true, it is not useful and did not help with determining the average starting salary of graduates.

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