
If the null-hypothesis is true, the P-value
(probability value) of a hypothesis test is the
probability of obtaining a sample statistic
with a value as extreme or more extreme
than the one determined from the sample
data (farther away).
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The smaller the P-Value, the more evidence
there is to reject the null hypothesis.
A very small P-Value indicates a rare
occurrence.
If the P-Value ≤ α, then you will reject the
null hypothesis.
If the P-Value > α, then you will fail to reject
the null hypothesis.
◦ If Ha contains <, the test is a left-tailed test.
 P is the area to the left of the test statistic.
◦ If Ha contains >, the test is a right-tailed test.
 P is the area to the right of the test statistic.
◦ If Ha contains ≠, the test is a two-tailed test.
 P is the area to the left of the negative test statistic,
and P is the area to the right of the positive test
statistic.
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Hypothesis tests for proportions occur (for
example) when a politician wants to know the
proportion of his or her constituents who favor a
certain bill or when a quality assurance engineer
tests the proportion of parts which are defective.
Z-Test for a Proportion P: A statistical test for a
population proportion P. It can be used when
np ≥ 5 and nq ≥ 5. (q is 1-p)
A test statistic is the sample proportion p-hat.
The standardized test statistic is z…formula to
follow later.
1.
2.
3.
4.
Verify that np ≥ 5 and nq ≥ 5. If these are
true, the distribution for p-hat will be
normal and you can continue; otherwise you
cannot use normal distribution for the
problem.
State the claim…Identify null and alternative
hypotheses.
Specify the level of significance (α).
Sketch the sampling distribution (make a
curve).
5.
6.
Determine any critical values (see next
slide). These will be borders between
rejection regions and non-rejection regions
(below). They will be the same values each
time.
Determine any rejection regions. These are
a range of values for which the Ho is not
probable. If a test statistic falls into this
region, Ho is rejected. A critical value
separates the rejection region from the
non-rejection region.
Tailed
Significance Level
Critical Value
Left
0.10
-1.28
Right
0.10
1.28
Two
0.10
±1.645
Left
0.05
-1.645
Right
0.05
1.645
Two
0.05
±1.96
Left
0.01
-2.33
Right
0.01
2.33
Two
0.01
±2.575
7.
8.
9.
Find the z-score (standard score):
Make a decision to
reject or fail to reject
Ho.
pˆ  p
z
pq
n
Interpret the decision
in the context of the original claim.
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A medical researcher claims that less than
20% of adults in the U.S. are allergic to a
medication. In a random sample of 100
adults, 15% say they have such an allergy. At
α = 0.01, is there enough evidence to
support the researcher’s claim?
n = 100, p = 0.20, q = 0.80
1. np = 20, nq = 80…you can continue.
2. Ho: p ≥ 0.2, Ha: p < 0.2
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Since Ha is <, this is a left-tailed test, and
since α = 0.01, we will be using the critical
value as -2.33 (they use the symbol zo for
this).
See drawing on board for sketch.
The rejection region is z < -2.33.
The standardized test statistic (z) is:
0.15  0.2
z
 1.25
(0.2)(0.8)
100
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Since z = -1.25, and this is not in the
rejection region, you should decide not to
reject the null hypothesis.
Interpretation: There is not enough evidence
to support the claim that less than 20% of
adults in the U.S. are allergic to the
medication.


USA Today reports that 5% of US adults have
seen an extraterrestrial being. You decide to
test this claim and ask a random sample of
250 U.S. adults whether they have ever seen
an extraterrestrial being. Of those surveyed
8% reply yes. At α = 0.01, is there enough
evidence to reject the claim?
Your group is to complete and document all
the steps to come to the final answer. This
will be turned in.
There
will be some
on Friday 