Transcript Chapter 4

Chapter 7
Inferences Based on a Single
Sample: Tests of Hypotheses
The Elements of a Test of
Hypothesis
7 elements:
1.
2.
3.
4.
5.
6.
7.
The Null hypothesis
The alternate, or research hypothesis
The test statistic
The rejection region
The assumptions
The Experiment and test statistic calculation
The Conclusion
The Elements of a Test of
Hypothesis
Does a manufacturer’s pipe meet building
code?
 is the mean breaking strength of the pipe
in pounds per linear foot
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
x  2400 x  2400
z

x
 n
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
The Elements of a Test of
Hypothesis
Assume that s is a good approximation of 
Sample of 60 taken, x  2460, s=200
Test statistic is
x  2400 2460 2400 60
z


 2.12
28.28
s n
200 50
Test statistic lies in rejection region, therefore we
reject H0 and accept Ha that the pipe meets
building code
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 H0 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
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:
a. One-tailed, lower tailed
Ha: <2400
b. One-tailed, upper tailed
Ha: >2400
c. 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, x = 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
x  12
Test statistic z 
x
Rejection region z < -2.575 or z > 2.575
(two-tailed)
Large-Sample Test of Hypothesis
about a Population Mean
CLT applies, therefore no assumptions
about population are needed
Solve
x 12 x 12 11.85 12 11.85 12  .15
z
x


n

 100

s 10

.5 10
 3
Since z falls in the rejection region, we
conclude that at .01 level of significance the
observed mean differs significantly from 12
Observed Significance Levels: pValues
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: pValues
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: pValues
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
Small-Sample Test of Hypothesis about 
One-Tailed Test
H 0:
H a:
  0
  0
Test Statistic:
(or Ha:
t  t
  0
)
x  0
t
s n
Rejection region:
(or
Two-Tailed Test
  0
H a:
  0
Test Statistic:
t  t
when Ha:
H 0:
x  0
t
s n
Rejection region:
  0
where t and t/2 are based on (n-1) degrees of freedom
t  t 2
Large-Sample Test of Hypothesis
about a Population Proportion
Large-Sample Test of Hypothesis about p
One-Tailed Test
H0:
p  p0
Ha:
p  p0
Two-Tailed Test
(or Ha:
Test Statistic:
p  p0
)
pˆ  p0
z
 pˆ
where, according to H0,
Rejection region:
(or z  z when
z   z
p  p0
H0:
p  p0
Ha:
p  p0
Test Statistic:
pˆ  p0
z
p
 pˆ  p0 q0 n
Rejection region:
and
q0  1  p0
z  z 2
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
p0  3 pˆ falls between 0 and 1
Tests of Hypothesis about a
Population Variance
Hypotheses about the variance use the ChiSquare distribution and statistic
n  1s
The quantity 
has a sampling
distribution that follows the
chi-square distribution
assuming the population the
sample is drawn from is
normally distributed.
2
2
Tests of Hypothesis about a
Population Variance
Test of Hypothesis about  2
One-Tailed Test
H 0:
Ha:
 2   02
 2   02
Test Statistic:
Two-Tailed Test
H0 :
(or Ha: 
  02 )
2


n

1
s
2 
 02
2
2
2



Rejection region:
 
2
2
2
2




(or
when Ha:    0
1
Ha :
 2   02
 2   02
Test Statistic:
2


n

1
s
2 
 02
2
2



Rejection region:

2
2
Or     
1 2 
2
where  02 is the hypothesized variance and the distribution of  is based
on (n-1) degrees of freedom
2