AP Statistics Section 11.1 B
More on Significance Tests
Conditions for Significance Tests
The three conditions that should be satisfied before we
conduct a hypothesis test about an unknown
population mean or proportion are the same as they
were for confidence intervals:
1. _______
SRS from the population of interest.
2. Distribution of x and p̂ must be ________________
approx. Normal
For x : _________________________________
population Normal or CLT (n  30)
For p̂ : ________________________
np  10 and n(1 - p)  10
3. _________________________
Independent observations
If sampling w/o replacement ___________
N  10n
Example 1: Check that the conditions from the
paramedic example in section 11.1 A are met.
SRS:
SRS of 400
Normality of x :
n  400 so CLT gives a dist. that is approx. Normal
Independence:
Calls without replacement so
pop. of all calls  10(400) or 4000
Test Statistics
A significance test uses data in the form of a test
statistic. The following principles apply to most tests:
(1) the test statistic compares the value of the
parameter as stated in the H
__0 to an estimate of the
parameter from the sample data.
(2) values of the estimate far from the parameter
value in the direction specified by the alternative
hypothesis give evidence _____________
against H 0
(3) to assess how far the estimate is from the
parameter, standardize the estimate.
In many common situations, the
test statistic has the form:
sample value - hypothesized value
test statistic = ----------------------------------------standard deviation of the sample dist.
6.48  6.7
z
2
400
Why z  ?
We know the population standard deviation.
Because the result is over two standard
deviations below the hypothesized mean
6.7, it gives good evidence that the mean
RT this year is not equal to 6.7 minutes, but
rather, less than 6.7 minutes.
The probability, computed assuming
__________,
H 0 is true that the observed sample
outcome would take a value as
extreme as or more extreme than that
actually observed is called the
__________
p - value of the test.
The smaller the P-value is, the
stronger the evidence is against
provided by the data.
Example 3: Let’s go back to our paramedic example.
The P-value is the probability of getting a sample
result at least as extreme as the one we did ( x = 6.48)
if H 0 :   67 were true. In other words, the P-value is
P( x  6.48) calculated assuming   6.7 . We just
found the z-score for this exact situation, so using
Table A or our calculator, this P-value is _______.
.0139 So if
H 0 is true, and the mean RT this year is still 6.7
minutes, there is about a _____
1.4% chance that the city
manager would obtain a sample of 400 calls with a
mean RT of 6.48 minutes or less. The small P-value
provides strong evidence against H 0 and in favor of
the alternative H a :   67 minutes.
If the Ha is two-sided, both
directions count when figuring the
P-value.
Example 4: Suppose we know that differences in job satisfaction scores
in Example 3 of section 11.1 A follow a Normal distribution with
standard deviation   60. If there is no difference in job satisfaction
between the two work environments, the mean is _______.
  0 Thus
H0: ________.
  0 The Ha says simply “there is a difference,” thus
Ha:________.
  0 Data from 18 workers gave 17. That is, these workers
preferred the self-paced environment on average. Find the p-value for
this situation and interpret it.
17  0
z
 1.20
60
18
.1151
P - value  2(.1151)  .2302
A p-value of .2302 indicates that 23.02% of
the time we will get a sample where x is at
least as big as 17 when   0 . An outcome
that would occur this often when   0 is
not good evidence that   0.
Statistical Significance
We can compare the P-value with a fixed value
that we regard as decisive. This amounts to
announcing in advance how much evidence
against H 0 we will insist on. The decisive value of
P is called the significance level. We write it as
____, the Greek letter alpha.

If the P-value   , we say that the data are
statistically significant at level 
Example 5: Back to the paramedic
example. We found the P = 0.0139. The
result is statistically significant at the  .05
.05 but it is not significant
level since P < ____
at the   .01 level since P > ____
.01
“Significant” in the statistical sense does
not mean “_____________.”
important It means
simply “not likely to happen just by
_________.”
chance
Interpreting Results in Context
The final step in performing a significance
test is to draw a conclusion about the
competing claims you were testing. As with
confidence intervals, your conclusion
should have a clear connection to your
calculations and should be stated in the
context of the problem. These are called
the 3 C’s.
In significance testing, there are
two accepted methods for drawing
conclusions:
In examples 3 and 4 of this section
we simply stated the p-value and
interpreted it in the context of the
problem.
In example 5, we went on to
determine if the data was statistically
significant be comparing our P-value to
our significance level . We can either
_______
reject or _______________
fail to reject the Ho
based on whether our result is
statistically significant at a given
significance level.
Warning: if you are going to draw
a conclusion based on statistical
significance, then the significance
level should be stated before the
data are produced.
Example 6: For the paramedic example, we
calculated the P-value to be 0.0139. If we
were using an   .05 significance level, we
would _____
________ )
reject H :   6.7 minutes ( conclusion
since ______
connection ). It appears
p  .05 ( __________
that the mean response time to all lifethreatening calls this year is less than last
year’s average of 6.7 minutes ( context
______ ).
0
Finally, stating a P-value is more informative
than simply giving a “reject” or “fail to reject”
conclusion at a given significance level. For
example, a P-value of 0.0139 allows us to
reject H 0 at the   .05level. But the P-value,
0.0139 gives us a better sense of how strong the
evidence against H 0 is. The P-value is the
smallest  level at which the data are significant.