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Population Parameters
In a study, we seek to gain information about the target population. There is a number of things
we can test about the population parameters, actual values. Two common ones are:
1) Population average, denoted by Greek Mu (“mew”),
2) Population percentage, denoted by Greek Pi (“pie”),
Unfortunately, we do not know the true values for and and realistically cannot, unless we
sample the entire population. We can only estimate them based on the sample we collect. The
values we collect from the sample are sample statistics and are estimators for the respective
population parameters. These estimators for the values above, respectively, are notated:
1) ̂ (“mew-hat”)
2) ̂ (“pie-hat”)
Example 1: Because of variation in the manufacturing process, tennis balls produced by a
particular machine do not have identical diameters. Let denote the true average diameter
for tennis balls currently being produced. Suppose that the machine was initially calibrated to
achieve the design specification in. However, the manufacturer is now concerned that
the diameters no longer conform to this specification. If sample evidence suggests that the
true average diameter for tennis balls is not 3 inches, the production process will have to be
halted while the machine is recalibrated. Because stopping the production is costly, the
manufacturer wants to be quite sure that the true average diameter is not 3 inches before
undertaking recalibration. What are the competing hypotheses
SOLUTION:
Under the original assumption, . The researcher wants to test whether . So:
Example 2: A long-used chemical in a particular carpet-cleaning product has been known to
successfully remove dark stains 70% of the time. After extensive research, the product's
formula is modified. The head of production must decide whether or not to sell the new
product. Write null and alternative hypotheses for conducting an experiment that might help
him decide.
SOLUTION:
Under original specifications, the proportion of time the product works is . He is
concerned that . If it is truly less effective, then he will not sell the new product. That is,
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