Download Section 6.2

Section 6.2
Introduction to
Hypothesis Testing
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Rejection Regions
Sampling distribution for
Rejection Region
z
z0
Critical Value z0
The rejection region is the range of values for which
the null hypothesis is not probable. It is always in the
direction of the alternative hypothesis. Its area is equal
to .
A critical value separates the rejection region from the
non-rejection region.
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Critical Values
The critical value z0 separates the rejection region from
the non-rejection region. The area of the rejection region
is .
Rejection
region
Rejection
region
z0
z0
Find z0 for a right-tail
test with = .05.
Find z0 for a left-tail
test with = .01.
z0 = –2.33
Rejection
region
Rejection
region
z0
z0
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Find –z0 and z0 for a two-tail test with
z0 = 1.645
–z0 = –2.575
and z0 = 2.575
= .01.
Using the Critical Value to Make Test Decisions
1. Write the null and alternative hypothesis.
Write H0 and Ha as mathematical statements.
Remember H0 always contains the = symbol.
2. State the level of significance.
This is the maximum probability of rejecting the null
hypothesis when it is actually true. (Making a type I error.)
3. Identify the sampling distribution.
The sampling distribution is the distribution for the test
statistic assuming that the equality condition in H0 is true
and that the experiment is repeated an infinite number of
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times.
5. Find the
rejection region.
4. Find the critical value.
Rejection Region
z0
6. Find the test statistic.
The critical value
separates the rejection
region of the sampling
distribution from the
non-rejection region.
The area of the critical
region is equal to the
level of significance of
the test.
Perform the calculations to standardize your sample statistic.
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7. Make your decision.
If the test statistic falls in the critical region, reject H0.
Otherwise, fail to reject H0.
8. Interpret your decision.
If the claim is the null hypothesis, you will either
reject the claim or determine there is not enough
evidence to reject the claim.
If the claim is the alternative hypothesis, you will
either support the claim or determine there is not
enough evidence to support the claim.
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The z-Test for a Mean
A cereal company claims the mean sodium content in one
serving of its cereal is no more than 230 mg. You work for a
national health service and are asked to test this claim. You
find that a random sample of 52 servings has a mean
sodium content of 232 mg and a standard deviation of 10
mg. At
= 0.05, do you have enough evidence to reject
the company’s claim?
1. Write the null and alternative hypothesis.
2. State the level of significance.
= 0.05
3. Determine the sampling distribution.
Since the sample size is at least 30, the sampling distribution is normal.
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Since Ha contains the > symbol, this is a right-tail test.
Rejection
region
z0
1.645
4. Find the critical value.
5. Find the rejection region.
6. Find the test statistic and standardize it.
n = 52
= 232 s = 10
7. Make your decision.
z = 1.44 does not fall in the rejection region, so fail to reject H0
8. Interpret your decision.
There is not enough evidence to reject the company’s claim that
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there is at most 230 mg of sodium in one serving of its cereal.
Using the P-value of a Test to Compare Areas
A
z
f
o
t
f
le
e
h 3
t
o 109
t
a 0.
re
= 0.05
z0 = –1.645
Rejection area
0.05
z0
z = –1.23
P = 0.1093
z
For a P-value decision, compare areas.
If
reject H0.
If
fail to reject H0.
For a critical value decision, decide if z is in the rejection region
If z is in the rejection region, reject H0. If z is not in the rejection
region, fail to reject H0.
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