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Part IV: Guesstimating and Hypothesizing with Confidence
Testing One Population Mean
When the variable is numerical (for example, age, income, time, and so on)
and only one population or group (such as all U.S. households or all col-
lege students) is being studied, you use the hypothesis test in this section
to examine or challenge a claim about the population mean. For example, a
child psychologist says that the average time that working mothers spend
talking to their children is 11 minutes per day, on average. (For dads, the
claim is 8 minutes.) The variable — time — is numerical, and the population
is all working mothers. Using statistical notation, μ represents the average
number of minutes per day that all working mothers spend talking to their
children, on average.
The null hypothesis is that the population mean, μ, is equal to a certain
claimed value, μ . The notation for the null hypothesis is H : μ = μ . So the
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null hypothesis in our example is H : μ = 11 minutes, and μ is 11. The three
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possibilities for the alternative hypothesis, H , are μ ≠ 11, μ < 11, or μ > 11,
a
depending on what you are trying to show. (See Chapter 14 for more on
alternative hypotheses.) If you suspect that the average time working moth-
ers spend talking with their kids is more than 11 minutes, your alternative
hypothesis would be H : μ > 11.
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To test the claim, you compare the mean you got from your sample ( ) with the
mean shown in H (μ ). To make a proper comparison, you look at the differ-
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ence between them, and divide by the standard error to take into account the
fact that your sample results will vary. (See Chapter 12 for all the info you need
on standard error.) This result is your test statistic. In the case of a hypothesis
test for the population mean, the test statistic turns out (under certain condi-
tions) to be a z-value (a value from the Z-distribution; see Chapter 9 ).
Then you can look up your test statistic on the appropriate table (in this
case, you look it up on the Z-table in the appendix), and find the chance that
this difference between your sample mean and the claimed population mean
really could have occurred if the claim were true.
The test statistic for testing one population mean (under certain conditions) is
where is the sample mean, σ is the population standard deviation (assume
for this case that this number is known), and z is a value on the Z-distribution.
To calculate the test statistic, do the following:
1. Calculate the sample mean, .
2. Find .
3. Calculate the standard error: .

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