Chapter 6 Statistical Inference 243
of textbooks. If we had used the z test statistic rather than the t statistic
in this example, the p value would have been 0.0455, and we would have
erroneously rejected the null hypothesis.
Constructing a t Confi dence Interval
Still we have a sample average that doesn’t completely match the adminis-
tration’s claim. Let’s construct a 95% confi dence interval for the mean value
to see in what range of values the true mean might lie. Because we don’t
know the value of s, we can’t use the confi dence interval equation discussed
earlier; we’ll have to use one based on the t distribution. The expression for
the t confi dence interval is
ax 2 t 1 2a/2, n 21
s
!n
, x 1 t 1 2a/2, n 21
s
!n
b
Here, t 1 2a/2, n 21 is the point on the t distribution with n 21 degrees of
freedom, such that the probability of a t random variable being less than it is
1 2a/2. To calculate this value in Excel, you use the TINV function. How-
ever, in the TINV function, you enter the value of a, not 12a/2. For ex-
ample, to calculate the value of t 1 2a/2, n 21 , enter the function TINV(a, n 2 1).
Use this information to construct a 95% confi dence interval.
To construct a 95% confi dence interval for the price of textbooks:
1 In cell A2, type 52202 TINV(0.05,24)*50/SQRT(25) and press Ta b.
2 In cell B2, type 52201 TINV(0.05,24)*50/SQRT(25) and press Enter.
The 95% confi dence interval is (199.36, 240.64). We do not expect the mean
price of textbooks to be much less than $200, nor should it be much greater
than $240. By comparison, the confi dence interval based on the standard nor-
mal distribution is (200.40, 239.60), so the confi dence intervals are very close
in size. Notice that the t confi dence interval includes 200, which means that
200 is not ruled out by the data, in agreement with our hypothesis test. This
agreement is not a coincidence, as can be shown with a little algebra.
The Robustness of t
When you use the t distribution to analyze your data, you’re assuming that
the data follow a normal distribution. What are the consequences if this turns
out not to be the case? The t distribution has a property called robustness,
which means that even if the assumption of normality is moderately violated,
