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Chapter
7
Elementary Statistics
Larson Farber
Hypothesis Testing
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A Statistical Hypothesis
Alternative
hypothesis Hacontains a statement
of inequality, such as.
Null hypothesis H0Statistical hypothesis
hat contains a
tatement of equality,uch as , = or .
If I am false,you are true
If I am false,
you are true
H0Ha
Complementary Statements
A claim about a population.
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Write the claim about the population. Then,write its complement. Either hypothesis, the
ull or the alternative, can represent the claim
A hospital claims its ambulance responsetime is less than 10 minutes.
H0 : 10 min
Ha : 10 min claim
Ha :60.0
p
H0 : 60.0p claim
Writing Hypotheses
A consumer magazine claims theproportion of cell phone calls made during
evenings and weekends is at most 60%.
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A type I error:Null hypothesis is actually
rue but the decision is to reject it.
Level of significance,
Maximum probability of committing
a type I error.
Decisi
on
Actual Truth of H0
Errors and Level ofSignificance
H0 True H0 False
Do not
reject H0
Reject H0
Correct
Decision
Correct
Decision
Type II
Error
Type I
Error
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Sampling distribution for x
The rejection regionis 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 equalto a.
Rejection Region
0z z0
A critical valueseparates the rejection
region from the non-rejection region
Critical Value z0
Rejection Regions
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z00
-z0
0 z0
Right-tail test Ha:>valu
Reject H0if z > z0otherwise fail to reject H0.
Two-tail testHa:value
Reject H0ifz >z0 or z
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Claim is H0
There is not
enoughevidence to
reject the claim
There is
enoughevidence to
reject the claim
Interpreting the Decision
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Claim is Ha
There is not
enoughevidence to
support the
claim
There is
enoughevidence to
support the
claim
Interpreting the Decision
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1. Write the null and alternative hypothesi
2. State the levelof significance
3. Identify the sampling distribution
Write H0 and Haas mathematical statements.
Remember H0 always contains the = symbol.
This is the maximum probability of rejecting the
null hypothesis when it is actually true. (Makinga type I error.)
The sampling distribution is the distribution for
the test statistic assuming that H0 is true and
that the experiment is repeated an infinite
number of times.
8 Steps in a Hypothesis Test
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7. Make your decision
6.Find the test statistic
5.Find therejection region
4. Find thecritical value
8. Interpret your decision
The critical value separates the rejection region
of the sampling distribution from the non-
rejection region.
Perform the calculations to standardize your
sample statistic.
If the test statistic falls in the critical region,reject H0. Otherwise, fail to reject H0.
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The critical value z0separates the rejection region
rom the non-rejection region. The area of the
ejection region is equal to a.
z0 0
ejection
egion
z00
Rejection
region
-z0 0 z0
Rejection
region
Rejection
region
ind z0 for a left-tail
est with a=.01Find z0 for a right-tail
test with a=.05
Find - z0 and z
0for a two-tail test with a=.01
z0=-2.33
-z0=-2.575 and z0 =2.575
z0=1.645
Critical Values
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A cereal company claims the meansodium content in one serving of its cereal is
no more than 230 mg. You work fora
national health service and are asked to test
this claim. You find that a random sample of52 servings has a mean sodium content of
232 milligrams and a standard deviation of
10 mg. At a= 0.05, do you have enough
evidence to reject the companys claim?1. Write the null and alternative hypothesis
H0: 230 mg.(claim) Ha: > 230 mg.
2. State the level of significance
a= 0.05
3. Determine the sampling distribution
Since the sample size is at least 30, the sampling
distribution is normal.
The z-test for a Mean
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7. Make your decision
6. Find the test statistic
8. Interpret your decision
5. Find the rejection
region
Rejection
region
Since Hacontains the > symbol, this is a right tail tes
n=52
x = 232s=10
44.1
387.1
2
52
10
230232
z
= 1.44 does not fall in the rejection region, soail to reject H0
There is not enough evidence to reject the
ompanys claim that there is at most 230mg of
z00
1.645
4. Find the critical
value