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AP Statistics Section 9.1 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 approximately
Normal
x
For : _________________________________
population Normal or CLT (n  30)
p̂
For : ________________________
np  10 and n(1 - p)  10
Independent observations
3. _________________________
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 x 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 sample statistic 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
 2.2
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.
P-values
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
H 0 provided by the data.
I suggest the following format when interpreting a pvalue:
parameter is ________,
Assuming the population ________
null value there
is a ________________%
P  value as a percent chance of getting a sample
statistic as extreme as ___________.
sample statistic This is
little / mod erate / strong evidence that
_________________
_____________________________.
give a conclusion about the H a in context
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 :   6.7 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
Interpret this p-value.
normalcdf (10000,2.2,0,1)
Assuming the population mean is 6.7, there is a 1.39%
chance of getting a sample statistic as extreme as 6.48.
This is strong evidence that the mean response time is less
than 6.7 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 9.1 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 x 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
Assuming the mean of the population
differences of JDS scores is 0, there is a 23.02%
chance of getting a sample mean difference as
extreme as 17. This is very little evidence of a
difference between job satisfaction for machine
paced vs self paced.
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 upon. 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 level
since P < .05 but it is not significant 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
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.
If no significance level is given, you will simply
interpret the p-value in context as we just did in
examples 3 & 4.
If a significance level is given, we can either
_______or
reject ______________
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.
I suggest the following formats
when writing your conclusion to a
test with a significance level:
Example 6: Consider our paramedic
example again. If we conducted a .05
significance test, write a conclusion in
context.
Since the P - value of .0139 is less than our significance level of .05,
I will reject the H o in favor of the H a . I can conclude that the mean
of the paramedic response times is less than 6.7 minutes.
Example 7: Consider our paramedic example
once again. If we conducted a .01 significance
test, write a conclusion in context.
Since the P - value of .0139 is greater than our significance
level of .01, I fail to reject the H o . There is insufficient evidence
to conclude that the true mean of the paramedic response times
is less then 6.7 minutes.
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   .05 level. But the P-value,
0.0139 gives us a better sense of how strong the
evidence against H 0 is.