months for the results of our test of the bulbs’ lifetime. So, we run our test of 1,000 light
bulbs and determine the average simulated lifetime of the bulbs in months. We discover
the following:
Sample mean (written as
x
)
=
37.7 months
Standard deviation (written as
s
)
=
.42 months
We use a
t
test in this case to determine if the null hypothesis should be rejected or ac
cepted (FTR) because we do not know the population’s standard deviation. However, be
cause the sample size is so large, the
t
value is the same as the
Z
value.
A
t
test is constructed as follows:
t
=
Samplevalue–Hypothesized population value
Standa
rd deviationof thesamplevalue
In the Longlast light bulb case the calculation would be:
t
=

37 7
36 0
42
4 05
.
.
.
.
See Table 13.3 for the appropriate statistical tests for the different forms of hypotheses.
Step 5: Compare Results to the Null Hypothesis and Make
a Decision
Looking at a
t
distribution table we see that the
t
value for n1 degrees of freedom (999
in our case, but “infinity” on the table) and for an alpha (the level of significance,
which was .01 in our case) in a “onetail” test (on a normal curve, the area under the
curve at only one end of the curve rather than both ends) was 2.326. Since our
t
value
of 4.05 exceeded this 2.326 value, we can say we are
more than
99 percent confident
that our sample mean of 37.7 months is drawn from a population whose mean is more
than 36. Or, in other words, the chances that our sample mean would be 37.7 when
the actual population mean is 36 months is very, very unlikely (less than one in a hun
dred). Because the odds of the null hypothesis being true (that our population mean is
36 months or less) are so small, it is much more reasonable to assume that the reason
our sample mean was 37.7 months is that the null hypothesis should be rejected and
our alternative hypothesis should be accepted (that the population mean is greater
than 36 months).
Chapter 13
Advanced Data Analysis
229
In Step 1 we established a decision rule that said if we reject the null hypothesis and
accept the alternative hypothesis we will market the Longlast light bulb, so that is what
we should do. Referring to Table 13.2, this decision means we either will make the cor
rect decision (cell 4), or will be making a Type 1 error (cell 3). The actual chance that
we are making a Type 1 error is called the
“p
value,” which in this case would be the
likelihood of getting a sample mean of 37.7 months if the null hypothesis were true. This
is very remote (less than 1 chance in 100), which is very reassuring of the correctness
of our decision.
While our null hypothesis was stated as a test of means (i.e., Was our sample’s aver
age of 37.7 months statistically significantly different from the hypothesized population’s
average of 36 months?), we could state hypotheses as frequency distribution, differences
between subgroups (e.g., men versus women), or proportions (e.g., percentages of college
educated versus noncollege graduates intending to buy our product). Table 13.3 shows the
appropriate statistical tests for different forms of hypotheses. Once we set up the hypoth