Transcript Chapter 8

Statistics
Chapter 8: Inferences Based on a
Single Sample: Tests of Hypotheses
Where We’ve Been


Calculated point estimators of
population parameters
Used the sampling distribution of a
statistic to assess the reliability of an
estimate through a confidence interval
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
2
Where We’re Going


Test a specific value of a population
parameter
Measure the reliability of the test
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
3
8.1: The Elements of a Test of
Hypotheses
µ?
Confidence Interval
Where on the number line do the data point us?
(No prior idea about the value of the parameter.)
Hypothesis Test
Do the data point us to this particular value?
(We have a value in mind from the outset.)
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
µ?
µ 0?
4
8.1: The Elements of a Test of
Hypotheses
Null Hypothesis: H0
•This will be supported
unless the data
provide evidence that it
is false
• The status quo
Alternative Hypothesis: Ha
•This will be supported if
the data provide sufficient
evidence that it is true
• The research hypothesis
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
5
8.1: The Elements of a Test of
Hypotheses


If the test statistic has a high
probability when H0 is true, then H0 is
not rejected.
If the test statistic has a (very) low
probability when H0 is true, then H0 is
rejected.
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
6
8.1: The Elements of a Test of
Hypotheses
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
7
8.1: The Elements of a Test of
Hypotheses
Reality ↓ / Test Result → Do not reject H0
H0 is true
H0 is false
Correct!
Type II Error: not
rejecting a false null
hypothesis
P(Type II error) = β
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
Reject H0
Type I Error:
rejecting a true null
hypothesis
P(Type I error) = α
Correct!
8
8.1: The Elements of a Test of
Hypotheses
Note: Null hypotheses
are either rejected, or
else there is insufficient
evidence to reject them.
(I.e., we don’t accept
null hypotheses.)
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
9
8.1: The Elements of a Test of
Hypotheses
•
•
•
•
•
•
•
Null hypothesis (H0): A theory about the values of one or more parameters
•
Ex.:
H0: µ = µ0 (a specified value for µ)
Alternative hypothesis (Ha): Contradicts the null hypothesis
•
Ex.:
H0: µ ≠ µ0
Test Statistic: The sample statistic to be used to test the hypothesis
Rejection region: The values for the test statistic which lead to rejection of
the null hypothesis
Assumptions: Clear statements about any assumptions concerning the
target population
Experiment and calculation of test statistic: The appropriate calculation for
the test based on the sample data
Conclusion: Reject the null hypothesis (with possible Type I error) or do
not reject it (with possible Type II error)
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
10
8.1: The Elements of a Test of
Hypotheses
Suppose a new interpretation of the rules by
soccer referees is expected to increase the
number of yellow cards per game. The
average number of yellow cards per game
had been 4. A sample of 121 matches
produced an average of 4.7 yellow cards
per game, with a standard deviation of .5
cards. At the 5% significance level, has
there been a change in infractions called?
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
11
8.1: The Elements of a Test of
Hypotheses
H0: µ = 4
H a: µ ≠ 4
Sample statistic:
α= .05
= 4.7
Assume the sampling distribution is normal.
Test statistic:
x   0 4.7  4
z* 

 10.94
sx
.064
Conclusion: z.025 = 1.96. Since z* > z.025 , reject H0.
(That is, there do seem to be more yellow cards.)
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
12
8.2: Large-Sample Test of a
Hypothesis about a Population Mean
The null hypothesis is
usually stated as an
equality …
Ha: µ < µ0
H0: µ = µ0
Ha: µ ≠ µ0
Ha: µ > µ0
… even though the alternative hypothesis
can be either an equality or an inequality.
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
13
8.2: Large-Sample Test of a
Hypothesis about a Population Mean
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
14
8.2: Large-Sample Test of a
Hypothesis about a Population Mean
Rejection Regions for Common Values of α
Lower Tailed
Upper Tailed
Two tailed
α = .10
z < - 1.28
z > 1.28
| z | > 1.645
α = .05
z < - 1.645
z > 1.645
| z | > 1.96
α = .01
z < - 2.33
z > 2.33
| z | > 2.575
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
15
8.2: Large-Sample Test of a
Hypothesis about a Population Mean
One-Tailed Test
 H0 : µ = µ0
Ha : µ < or > µ0
x  0
Test Statistic: z 
x
Rejection Region: | z | > zα
Conditions:
Two-Tailed Test
 H0 : µ = µ0
Ha : µ ≠ µ0
x  0
Test Statistic: z 
x
Rejection Region: | z | > zα/2
1) A random sample is selected from the target population.
2) The sample size n is large.
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
16
8.2: Large-Sample Test of a
Hypothesis about a Population Mean

The Economics of Education Review
(Vol. 21, 2002) reported a mean salary
for males with postgraduate degrees of
$61,340, with an estimated standard
error (s ) equal to $2,185. We wish to
test, at the α = .05 level, H0: µ =
$60,000.
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
17
8.2: Large-Sample Test of a
Hypothesis about a Population Mean
The Economics of
Education Review (Vol.
21, 2002) reported a
mean salary for males
with postgraduate
degrees of $61,340,
with an estimated
standard error (s )
equal to $2,185. We
wish to test, at the
α
= .05 level,
H0: µ
= $60,000.

H0 : µ = 60,000
Ha : µ ≠ 60,000
Test Statistic:
z
x  0
x
61,340  60,000
z
2,185
z  .613
Rejection Region: | z | > z.025 = 1.96
Do not reject H0
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
18
8.3:Observed Significance Levels: p Values
Suppose z = 2.12.
P(z > 2.12) = .0170.
Reject H0 at the α= .05 level
Do not reject H0 at the α= .01 level
But it’s pretty close, isn’t it?
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
19
8.3:Observed Significance Levels: p Values
The observed significance level, or p-value, for
a test is the probability of observing the results
actually observed (z*) assuming the null
hypothesis is true.
P( z  z* | H 0 )
The lower this probability, the less likely H0 is true.
McClave, Statistics, 11th ed. Chapter 8: Inferences Based
on a Single Sample: Tests of Hypotheses
20
8.3:Observed Significance Levels: p Values
Let’s go back to the
Economics of
Education Review
report ( = $61,340,
s = $2,185). This
time we’ll test
H0: µ = $65,000.
H0 : µ = 65,000
Ha : µ ≠ 65,000
Test Statistic:
z
x  0
x
61,340  65,000
2,185
z  1.675
z
p-value: P(|z| > 1.675) = .047 * 2
=0.094
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
21
8.3:Observed Significance Levels: p Values

Reporting test results


Choose the maximum tolerable value of α
If the p-value ≤ α, reject H0
If the p-value > α, do not reject H0
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
22
8.4: Small-Sample Test of a
Hypothesis about a Population Mean
If the sample is small and  is unknown,
testing hypotheses about µ requires the
t-distribution instead of the z-distribution.
x  0
t
s/ n
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
23
8.4: Small-Sample Test of a
Hypothesis about a Population Mean
One-Tailed Test
 H0 : µ = µ0
Ha : µ < or > µ0
Two-Tailed Test
 H0 : µ = µ0
Ha : µ ≠ µ0
x  0
Test Statistic: t 
s/ n
x  0
Test Statistic: t 
s/ n
Rejection Region: | t | > tα
Rejection Region: | t | > tα/2
Conditions:
1) A random sample is selected from the target population.
2) The population from which the sample is selected is
approximately normal.
3) The value of tα is based on (n – 1) degrees of freedom
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
24
8.4: Small-Sample Test of a
Hypothesis about a Population Mean
Suppose copiers average 100,000
between paper jams. A salesman
claims his are better, and offers to
leave 5 units for testing. The average
number of copies between jams is
100,987, with a standard deviation of
157. Does his claim seem believable?
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
25
8.4: Small-Sample Test of a
Hypothesis about a Population Mean
Suppose copiers
average 100,000
between paper jams. A
salesman claims his are
better, and offers to
leave 5 units for testing.
The average number of
copies between jams is
100,987, with a standard
deviation of 157. Does
his claim seem
believable?
H0 : µ = 100,000
Ha : µ > 100,000
Test Statistic:
x  0
t
s/ n
100,987  100,000
t
157 / 5
t  14.06
p-value: P(tdf=4 > 14.06) < .0005
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
26
8.4: Small-Sample Test of a
Hypothesis about a Population Mean
Suppose copiers
average 100,000
between paper jams. A
salesman claims his are
better, and offers to
leave 5 units for testing.
The average number of
copies between jams is
100,987, with a standard
deviation of 157. Does
his claim seem
believable?
HReject
the null hypothesis
0 : µ = 100,000
on the very low
Hbased
a : µ > 100,000
probability
Test
Statistic:of seeing the
observed results if the null
were true.
x  0
 claim does seem
So,t the
s/ n
plausible.
100,987  100,000
t
157 / 5
t  14.06
p-value: P(|tdf=4| > 14.06) < .0005
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
27
8.5: Large-Sample Test of a Hypothesis about
a Population Proportion
One-Tailed Test
 H0 : p = p0
Ha : p < or > p0
pˆ  p0
Test Statistic: z 
 pˆ
Two-Tailed Test
 H0 : p = p0
Ha : p ≠ p0
pˆ  p0
Test Statistic: z 
Rejection Region: | z | > zα
p0 = hypothesized value of p,  pˆ 
 pˆ
Rejection Region: | z | > zα/2
p0 q0
, and q0 = 1 - p0
n
Conditions: 1) A random sample is selected from a binomial population.
2) The sample size n is large (i.e., np0 and nq0 are both ≥ 15).
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
28
8.5: Large-Sample Test of a Hypothesis about
a Population Proportion
Rope designed for use in
the theatre must
withstand unusual
stresses. Assume a
brand of 3” three-strand
rope is expected to have
a breaking strength of
1400 lbs. A vendor
receives a shipment of
rope and needs to
(destructively) test it.
The vendor will reject
any shipment which
cannot pass a 1% defect
test (that’s harsh, but so
is falling scenery during
an aria). 1500 sections
of rope are tested, with
20 pieces failing the test.
At the α = .01 level,
should the shipment be
rejected?
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
29
8.5: Large-Sample Test of a Hypothesis about
a Population Proportion
The vendor will reject
any shipment that cannot
pass a 1% defects test .
1500 sections of rope
are tested, with 20
pieces failing the test. At
the α = .01 level, should
the shipment be
rejected?
H0: p = .01
Ha: p > .01
Rejection region: |z| > 2.33
Test statistic:
z
pˆ  p0
 pˆ
.013  .01
z
(.013)(. 987) / 1500
z  1.14
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
30
8.5: Large-Sample Test of a Hypothesis about
a Population Proportion
The vendor will reject
any shipment that cannot
pass a 1% defects test .
1500 sections of rope
are tested, with 20
pieces failing the test. At
the α = .01 level, should
the shipment be
rejected?
H0: p = .01
Ha: p > .01
There is insufficient
Rejection
region:
|z| >
evidence
to reject
the2.33
null hypothesis based
Test statistic:
on the sample results.
z
pˆ  p0
 pˆ
.013  .01
z
(.013)(. 987) / 1500
z  1.14
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
31
8.6: Calculating Type II Error Probabilities:
More about β
To calculate P(Type II), or β, …
1. Calculate the value(s) of that divide the “do not reject”
region from the “reject” region(s).
 s 

Upper-tailed test: x0   0  z  x   0  z 
 n
 s 
Lower-tailed test: x0   0  z  x  0  z 

 n
Two-tailed test:

x0 L  0  z / 2 x   0  z / 2 


x0U   0  z / 2 x   0  z / 2 

McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
s 

n
s 

n
32
8.6: Calculating Type II Error Probabilities:
More about β
To calculate P(Type II), or β, …
1. Calculate the value(s) of that divide the “do not
reject” region from the “reject” region(s).
2. Calculate the z-value of 0 assuming the
alternative hypothesis mean is the true µ:
z
x0   a
x
The probability of getting this z-value for the
acceptance region is β.
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
33
8.6: Calculating Type II Error Probabilities:
More about β

The power of a test is the probability
that the test will correctly lead to the
rejection of the null hypothesis for a
particular value of µ in the alternative
hypothesis. The power of a test is
calculated as (1 - β ).
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
34
8.6: Calculating Type II Error Probabilities:
More about β
The Economics of
Education Review (Vol.
21, 2002) reported a mean
salary for males with
postgraduate degrees of
$61,340, with an
estimated standard error
(s ) equal to $2,185. We
wish to test, at the α = .05
level, H0: µ = $60,000.

H0 : µ = 60,000
Ha : µ ≠ 60,000
Test Statistic: z = .613;
zα/2=.025 = 1.96
We did not reject this null
hypothesis earlier, but what if
the true mean were
$62,000?
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
35
8.6: Calculating Type II Error Probabilities:
More about β
The Economics of
Education Review (Vol.
21, 2002) reported a mean
salary for males with
postgraduate degrees of
$61,340, with s equal to
$2,185.
We did not reject this null
hypothesis earlier, but
what if the true mean were
$62,000?
xO,L= 60,000 - 1.96*2,185 =
55,717.4
xO,U= 60,000 +1.96*2,185 =
64,282.6
ZL = (55,717.4 –
62,000)/2,185 = -2.875
ZU = (64,282.6 62,000)/2,185 = 1.045
β = P(-2.875≤Z≤1.045) =
0.8511
The power of this test is 1 –
β = 1 – 0.8511 = 0.1489
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
36
8.6: Calculating Type II Error Probabilities:
More about β



For fixed n and α, the value of β decreases
and the power increases as the distance
between µ0 and µa increases.
For fixed n, µ0 and µa, the value of β
increases and the power decreases as the
value of α is decreased.
For fixed α, µ0 and µa, the value of β
decreases and the power increases as n is
increased.
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
37
8.6: Calculating Type II Error Probabilities:
More about β
For fixed n
and 
For fixed n, µ0
and µa
For fixed , µ0
and µa
 decreases
and (1-)
increases
 increases
and (1 - )
decreases
 decreases
and (1 -  )
increases
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
as |µ0 - µa|
increases
as 
decreases
as n
increases
38
8.7: Tests of Hypotheses about a
Population Variance
Random Variable X
Sample Statistic:
or p-hat
Sample Statistic:
s2
Hypothesis Test on µ or p
Based on z
Based on t
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
Hypothesis Test
on 2 or 
Based on
2
39
8.7: Tests of Hypotheses about a
Population Variance
The chi-square distribution is really a family of distributions,
depending on the number of degrees of freedom.
But, the population must be normally distributed for the
hypothesis tests on 2 (or ) to be reliable!
2 
(n  1) s 2
2
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
40
8.7: Tests of Hypotheses about a
Population Variance
One-Tailed Test
H 0 :  2   02
Two-Tailed Test

H a :  2   02  2   02

H 0 :  2   02
H a :  2   02
Test statistic:
2
(
n

1
)
s
2 
2
Test statistic:
2
(
n

1
)
s
2 
2
Rejection region:
Rejection region:
0
 2  12   2  2
0
 2  12 / 2 or  2  2 / 2
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
41
8.7: Tests of Hypotheses about a
Population Variance
Conditions Required for a Valid
Large- Sample Hypothesis Test for 2
1. A random sample is selected from the
target population.
2. The population from which the sample is
selected is approximately normal.
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
42
8.7: Tests of Hypotheses about a
Population Variance
Earlier, we considered the average number
of copies between jams for a brand of
copiers. The salesman also claims his
copiers are more predictable, in that the
standard deviation of jams is 125. In the
sample of 5 copiers, that sample standard
deviation was 157. Does his claim seem
believable, at the α = .10 level?
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
43
8.7: Tests of Hypotheses about a
Population Variance
Earlier, we considered the
average number of copies
between jams for a brand of
copiers. The salesman also
claims his copiers are more
predictable, in that the
standard deviation of jams is
125. In the sample of 5
copiers, that sample standard
deviation was 157. Does his
claim seem believable, at the
α = .10 level?
Two-Tailed Test
H 0 :  2  125
H a :  2  125
Test statistic:
2
(
5

1
)(
157
)
2 
 6.31
2
125
Rejection criterion:
 2  6.31  .205  9.48773
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
44
8.7: Tests of Hypotheses about a
Population Variance
Earlier, we considered the
average number of copies
between jams for a brand of
copiers.
The salesman
Do not
reject also
claims his copiers are more
the
null
reliable, in that the standard
hypothesis.
deviation
of jams is 125. In
the sample of 5 copiers, that
sample standard deviation
was 157. Does his claim
seem believable, at the α =
.10 level?
Two-Tailed Test
H 0 :  2  125
H a :  2  125
Test statistic:
2
(
5

1
)(
157
)
2 
 6.31
2
125
Rejection criterion:
 2  6.31  .205  9.48773
McClave, Statistics, 11th ed. Chapter 8: Inferences
Based on a Single Sample: Tests of Hypotheses
45