chapter 8 inferences based on a single sample: tests of hypothesis
TRANSCRIPT
Chapter 8
Inferences Based on a Single Sample: Tests of Hypothesis
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The Elements of a Test of Hypothesis
7 elements1. The Null hypothesis
2. The alternate, or research hypothesis
3. The test statistic
4. The rejection region
5. The assumptions
6. The Experiment and test statistic calculation
7. The Conclusion
3
The Elements of a Test of Hypothesis
Does a manufacturer’s pipe meet building code?
Null hypothesis – Pipe does not meet code
(H0): < 2400
Alternate hypothesis – Pipe meets specifications
(Ha): > 2400
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The Elements of a Test of Hypothesis
Test statistic to be used
Rejection region Determined by Type I error, which is the probability of rejecting the null hypothesis when it is true, which is . Here, we set =.05
Region is z>1.645, from z value table
n
xxz
x 24002400
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The Elements of a Test of Hypothesis
Assume that s is a good approximation of Sample of 60 taken, , s=200
Test statistic is
Test statistic lies in rejection region, therefore we reject H0 and accept Ha that the pipe meets building code
12.228.28
60
50200
240024602400
ns
xz
2460x
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The Elements of a Test of Hypothesis
Type I vs Type II Error
Conclusions and Consequences for a Test of Hypothesis
True State of Nature
Conclusion H0 True Ha True
Accept H0
(Assume H0 True)
Correct decision Type II error (probability )
Reject H0
(Assume Ha True)
Type I error (probability )
Correct decision
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The Elements of a Test of Hypothesis
1. The Null hypothesis – the status quo. What we will accept unless proven otherwise. Stated as H0: parameter = value
2. The Alternative (research) hypothesis (Ha) – theory that contradicts H0. Will be accepted if there is evidence to establish its truth
3. Test Statistic – sample statistic used to determine whether or not to reject Ho and accept Ha
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The Elements of a Test of Hypothesis
4. The rejection region – the region that will lead to H0 being rejected and Ha accepted. Set to minimize the likelihood of a Type I error
5. The assumptions – clear statements about the population being sampled
6. The Experiment and test statistic calculation – performance of sampling and calculation of value of test statistic
7. The Conclusion – decision to (not) reject H0, based on a comparison of test statistic to rejection region
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Large-Sample Test of Hypothesis about a Population Mean
Null hypothesis is the status quo, expressed in one of three forms
H0: = 2400
H0: ≤ 2400
H0: ≥ 2400
It represents what must be accepted if the alternative hypothesis is not accepted as a result of the hypothesis test
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Large-Sample Test of Hypothesis about a Population Mean
Alternative hypothesis can take one of 3 forms:
One-tailed, upper tail Ha: <2400
One-tailed, upper tail Ha: >2400
Two-tailed Ha: 2400
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Large-Sample Test of Hypothesis about a Population Mean
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Large-Sample Test of Hypothesis about a Population Mean
If we have: n=100, = 11.85, s = .5, and we want to test if 12 with a 99% confidence level, our setup would be as follows:
H0: = 12
Ha: 12
Test statistic
Rejection region z < -2.575 or z > 2.575 (two-tailed)
x
x
xz
12
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Large-Sample Test of Hypothesis about a Population Mean
CLT applies, therefore no assumptions about population are needed
Solve
Since z falls in the rejection region, we conclude that at .01 level of significance the observed mean differs significantly from 12
3.105.
15.
10
1285.11
100
1285.111212
sn
xxz
x
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Observed Significance Levels: p-Values
The p-value, or observed significance level, is the smallest that can be set that will result in the research hypothesis being accepted.
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Observed Significance Levels: p-Values
Steps:
Determine value of test statistic z
The p-value is the area to the right of z if Ha is one-tailed, upper tailed
The p-value is the area to the left of z if Ha is one-tailed, lower tailed
The p-valued is twice the tail area beyond z if Ha is two-tailed.
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Observed Significance Levels: p-Values
When p-values are used, results are reported by setting the maximum you are willing to tolerate, and comparing p-value to that to reject or not reject H0
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Small-Sample Test of Hypothesis about a Population Mean
When sample size is small (<30) we use a different sampling distribution for determining the rejection region and we calculate a different test statistic
The t-statistic and t distribution are used in cases of a small sample test of hypothesis about All steps of the test are the same, and an assumption about the population distribution is now necessary, since CLT does not apply
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Small-Sample Test of Hypothesis about a Population Mean
where t and t/2 are based on (n-1) degrees of freedom
Rejection region:Rejection region:
(or when Ha:
Test Statistic:Test Statistic:
Ha:Ha: (or Ha: )
H0:H0:
Two-Tailed TestOne-Tailed Test
Small-Sample Test of Hypothesis about
0
0
ns
xt 0
0 0
ns
xt 0
tt
tt 2tt
0
0
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Large-Sample Test of Hypothesis about a Population Proportion
Rejection region:Rejection region:
(or when
where, according to H0, and
Test Statistic:Test Statistic:
Ha:Ha: (or Ha: )
H0:H0:
Two-Tailed TestOne-Tailed Test
Large-Sample Test of Hypothesis about
0pp
p
0pp
p
ppz
ˆ
0ˆ
0pp
0pp
p
ppz
0ˆ
zz zz 2zz
0pp
0pp
nqpp 00ˆ 00 1 pq
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Large-Sample Test of Hypothesis about a Population Proportion
Assumptions needed for a Valid Large-Sample Test of Hypothesis for p•A random sample is selected from a binomial population•The sample size n is large (condition satisfied if falls between 0 and 1 pp ˆ0 3
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Calculating Type II Error Probabilities: More about
Type II error is associated with , which is the probability that we will accept H0 when Ha is true
Calculating a value for can only be done if we assume a true value for There is a different value of for every value of
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Calculating Type II Error Probabilities: More about
Steps for calculating for a Large-Sample Test about 1. Calculate the value(s) of corresponding to the
borders of the rejection region using one of the following:
Upper-tailed test:
Lower-tailed test:
Two-tailed test:
x
n
szzx
x 000
n
szzx
x 000
n
szzx
xL 000
n
szzx
xU 000
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Calculating Type II Error Probabilities: More about
2. Specify the value of in Ha for which is to be calculated.
3. Convert border values of to z values using the mean , and the formula
4. Sketch the alternate distribution, shade the area in the acceptance region and use the z statistics and table to find the shaded area,
a
x
axz
0
0xa
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Calculating Type II Error Probabilities: More about
The Power of a test – the probability that the test will correctly lead to the rejection of H0
for a particular value of in Ha. Power is calculated as 1- .
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Tests of Hypothesis about a Population Variance
Hypotheses about the variance use the Chi-Square distribution and statistic
The quantity has a sampling distribution that follows the chi-square distribution assuming the population the sample is drawn from is normally distributed.
2
21
sn
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Tests of Hypothesis about a Population Variance
where is the hypothesized variance and the distribution of is based on (n-1) degrees of freedom
Rejection region:
Or
Rejection region:
(or when Ha:
Test Statistic:Test Statistic:
Ha:Ha: (or Ha: )
H0:H0:
Two-Tailed TestOne-Tailed Test
Small-Sample Test of Hypothesis about
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2
2
20
2
20
22 1
sn
1
22
20
2 20
2 20
2
20
22 1
sn
22 21
22
2
22 2
02
220