Section 7.1 Introduction to Hypothesis Testing Larson/Farber 4th ed.
Hypothesis Testing Betsy Farber PPG_2
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Transcript of Hypothesis Testing Betsy Farber PPG_2
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Chapter
7
Elementary Statistics
Larson Farber
Hypothesis Testing
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A Statistical Hypothesis
Alternative
hypothesis Hacontains a statement
ofinequality, such as.
Null hypothesis H0Statistical hypothesis
hat contains a
tatement ofequality,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, aMaximum 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 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 equalto a.
Rejection Region
0z z0
A critical value separates 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 H0 ifz > z0otherwise fail to reject H0.
Two-tail testHa:value
Reject H0 ifz>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 level of significance
3. Identify the sampling distribution
Write H0 and Ha as 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 z0 separates 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 z0for 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 workfora
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 Ha contains the > symbol, this is a right tail tes
n=52
x = 232s=10
44.1387.1
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 ofodium in one serving of its cereal.
z00
1.645
4. Find the critical
value
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Find a c20 critical value for a left-tail test when
=17 and a = 0.05.
2 is the test statistic for the population variance. Its
ampling distribution is a c2 distribution with n-1 d.f.
Find critical values c20 for a two-tailed test when= 12 and a = 0.01.
The standardized test statistic is2
22 )1(
csn
c20 =7.962
c2l =2.603 and c2
R=26.757
0 1 0 2 0 3 0 4 0
Critical Values for 2
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A state school administrator says that the standard
deviation of test scores for 8th grade students who
took a life-science assessment test is less than 30.
You work for the administrator and are asked to test
this claim. You find that a random sample of 10 testshas a standard deviation of 28.8. At a = 0.01, do you
have enough evidence to support the administrators
claim? Assume test scores are normally distributed.
. Write the null and alternative hypothesis
H0
: 30 Ha: < 30 (claim)
. State the level of significance a= 0.01
3. Determine the sampling distribution
The sampling distribution is c2 with 10 - 1 = 9 d.f.
Test for
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7. Make your decision
6. Find the test statistic
8. Interpret your decision
n=10s = 28.8
c2 = 8.2944 does not fall in the rejection region,
so fail to reject H0
There is not enough evidence to support the
dministrators claim that the standard deviation is
2944.8
30
8.28)110()1(2
2
2
22
csn
0 1 0 2 0 3 0 4 0
5. Find the rejection
region
4. Find the critical
value
2.088