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Chapter 15: Commonly Used Hypothesis Tests: Formulas and Examples
4. Divide your result from Step 2 by the standard error found in Step 3.
The conditions for using this test statistic are that the population standard
deviation, σ, is known, and either the population has a normal distribution or
the sample size is large enough to use the CLT (n > 30); see Chapter 11.
For our example, suppose a random sample of 100 working mothers spend
an average of 11.5 minutes per day talking with their children. (Assume prior
research suggests the population standard deviation is 2.3 minutes.)

1. We are given that is 11.5, n = 100, and σ is 2.3.

2. Take 11.5 – 11 = +0.5.
3. Take 2.3 divided by the square root of 100 (which is 10) to get 0.23 for

the standard error.
4. Divide +0.5 by 0.23 to get 2.17. That’s your test statistic, which means your

sample mean is 2.17 standard errors above the claimed population mean.
The big idea of a hypothesis test is to challenge the claim that’s being made 229
about the population (in this case, the population mean); that claim is shown
in the null hypothesis, H . If you have enough evidence from your sample
o
against the claim, H is rejected.
o
To decide whether you have enough evidence to reject H , calculate the
o
p-value by looking up your test statistic (in this case 2.17) on the standard
normal (Z-) distribution — see the Z-table in the appendix — and take 1 minus
the probability shown. (You subtract from 1 because your H is a greater-
a
than hypothesis and the table shows less-than probabilities.)
For this example you look up the test statistic (2.17) on the Z-table and find
the (less-than) probability is 0.9850, so the p-value is 1 – 0.9850 = 0.015. It’s
quite a bit less than your (typical) significance level 0.05, which means your
sample results would be considered unusual if the claim (of 11 minutes) was
true. So reject the claim (H : μ = 11 minutes). Your results support the alter-
o
native hypothesis H : μ > 11. According to your data, the child psychologist’s
a
claim of 11 minutes per day is too low; the actual average is greater than that.
For information on how to calculate p-values for the less-than or not-equal-to
alternatives, also see Chapter 14.
Handling Small Samples and Unknown
Standard Deviations: The t-Test
In two cases, you can’t use the Z-distribution for a test statistic for one popu-
lation mean. The first case is where the sample size is small (and by small,

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