chapter 8 - wessa.net · different sampling distribution for determining the rejection region and...
TRANSCRIPT
Chapter 8
Inferences Based on a Single
Sample: Tests of Hypothesis
The Elements of a Test of
Hypothesis
7 elements
1. 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
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
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
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
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
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
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
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
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
Large-Sample Test of Hypothesis
about a Population Mean
Rejection Regions for Common Values of
Alternative Hypotheses
Lower-Tailed Upper-Tailed Two-Tailed
= .10 z < -1.28 z > 1.28 z < -1.645 or z > 1.645
= .05 z < -1.645 z > 1.645 Z < -1.96 or z > 1.96
= .01 z < -2.33 z > 2.33 Z < -2.575 or z > 2.575
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
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
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.
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.
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
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
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
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
001 pq
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
ˆ03
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
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
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
ax
z
0
0x
a
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- .
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
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
2
0
2
2
2
0
2
2
0
2
2 1
sn
1
22
2
0
2
2
0
2
2
0
2
2
0
2
2 1
sn
22
21
22
2
22
2
0
2
22
0