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Part IV: Guesstimating and Hypothesizing with Confidence
I mean dropping below 30 or so) the second case is when the population
standard deviation, σ, is not known, and you have to estimate it using the
sample standard deviation, s. In both cases, you have less reliable informa-
tion on which to base your conclusions, so you have to pay a penalty for this
by using a distribution with more variability in the tails than a Z-distribution
has. Enter the t-distribution. (See Chapter 10 for all things t-distribution,
including its relationship with the Z.)
A hypothesis test for a population mean that involves the t -distribution is
called a t-test. The formula for the test statistic in this case is:
, where t is a value from the t-distribution with n–1 degrees
n-1

of freedom.
Note it is just like the test statistic for the large sample and/or normal distri-
bution case (see the section “Testing One Population Mean”), except σ is not
known, so you substitute the sample standard deviation, s, instead, and use a
t-value rather than a z-value.
Because the t-distribution has fatter tails than the Z-distribution, you get a
larger p-value from the t-distribution than one that the standard normal (Z-)
distribution would have given you for the same test statistic. A bigger p-value
means less chance of rejecting H . Having less data and/or not knowing the
o
population standard deviation should create a higher burden of proof.
Putting the t-test to work
Suppose a delivery company claims they deliver their packages in 2 days on
average, and you suspect it’s longer than that. The hypotheses are H : μ = 2
o
versus H : μ > 2. To test this claim, you take a random sample of 10 packages
a
and record their delivery times. You find the sample mean is days,
and the sample standard deviation is 0.35 days. (Because the population
standard deviation, σ, is unknown, you estimate it with s, the sample stan-
dard deviation.) This is a job for the t-test.
Because the sample size is small (n =10 is much less than 30) and the popula-
tion standard deviation is not known, your test statistic has a t-distribution. Its
degrees of freedom is 10 – 1 = 9. The formula for the test statistic (referred to
as the t-value) is:
To calculate the p-value, you look in the row in the t-table (in the appendix)
for df = 9. Your test statistic (2.71) falls between two values in the row for
df = 9 in the t-table: 2.26 and 2.82 (rounding to two decimal places). To calculate

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