Download Power Curves and OC Curves

9
Chapter
One-Sample Hypothesis
Testing
Logic of Hypothesis Testing
Statistical Hypothesis Testing
Testing a Mean: Known Population
Variance
Testing a Mean: Unknown Population
Variance
Testing a Proportion
Power Curves and OC Curves
(Optional)
Tests for One Variance (Optional)
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc.
Logic of Hypothesis Testing
•
9-2
Steps in Hypothesis Testing
Step 1: State the assumption to be tested
Step 2: Specify the decision rule
Step 3: Collect the data to test the
hypothesis
Step 4: Make a decision
Step 5: Take action based on the decision
Logic of Hypothesis Testing
State the Hypothesis
•
•
•
•
9-3
Hypotheses are a pair of mutually
exclusive, collectively exhaustive
statements about the world.
One statement or the other must be true,
but they cannot both be true.
H0: Null Hypothesis
H1: Alternative Hypothesis
These two statements are hypotheses
because the truth is unknown.
Logic of Hypothesis Testing
State the Hypothesis
•
•
•
•
9-4
Efforts will be made to reject the null
hypothesis.
If H0 is rejected, we tentatively conclude H1
to be the case.
H0 is sometimes called the maintained
hypothesis.
H1 is called the action alternative because
action may be required if we reject H0 in
favor of H1.
Logic of Hypothesis Testing
Can Hypotheses be Proved?
•
We cannot prove a null hypothesis, we can
only fail to reject it.
Role of Evidence
•
9-5
The null hypothesis is assumed true and a
contradiction is sought.
Logic of Hypothesis Testing
Types of Error
•
•
9-6
Type I error: Rejecting the null hypothesis
when it is true. This occurs with probability
a.
Type II error: Failure to reject the null
hypothesis when it is false. This occurs
with probability b.
Statistical Hypothesis Testing
•
•
A statistical hypothesis is a statement about
the value of a population parameter q.
A hypothesis test is a decision between two
competing mutually exclusive and
collectively exhaustive hypotheses about
the value of q.
Left-Tailed Test
9-7
Two-Tailed Test
Right-Tailed Test
Statistical Hypothesis Testing
•
The direction of the test is indicated by H1:
> indicates a right-tailed test
< indicates a left-tailed test
≠ indicates a two-tailed test
9-8
Statistical Hypothesis Testing
When to use a One- or Two-Sided Test
•
•
•
9-9
A two-sided hypothesis test (i.e., q ≠ q0) is used
when direction (< or >) is of no interest to the
decision maker
A one-sided hypothesis test is used when
- the consequences of rejecting H0 are
asymmetric, or
- where one tail of the distribution is of special
importance to the researcher.
Rejection in a two-sided test guarantees
rejection in a one-sided test, other things
being equal.
Statistical Hypothesis Testing
Decision Rule
•
•
•
9-10
A test statistic shows how far the sample
estimate is from its expected value, in
terms of its own standard error.
The decision rule uses the known sampling
distribution of the test statistic to establish
the critical value that divides the sampling
distribution into two regions.
Reject H0 if the test statistic lies in the
rejection region.
Statistical Hypothesis Testing
Decision Rule for Two-Tailed Test
•
Reject H0 if the test statistic < left-tail
critical value or if the test statistic > righttail critical value.
Figure 9.2
- Critical value
9-11
+ Critical value
Statistical Hypothesis Testing
Decision Rule for Left-Tailed Test
•
Reject H0 if the test statistic < left-tail
critical value.
Figure 9.2
- Critical value
9-12
Statistical Hypothesis Testing
Decision Rule for Right-Tailed Test
•
Reject H0 if the test statistic > right-tail
critical value.
Figure 9.2
+ Critical value
9-13
Statistical Hypothesis Testing
Type I Error
•
a, the probability of a Type I error, is the level
of significance (i.e., the probability that the test
statistic falls in the rejection region even
though H0 is true).
a = P(reject H0 | H0 is true)
•
•
9-14
A Type I error is sometimes referred to as a
false positive.
For example, if we choose a = .05, we expect to
commit a Type I error about 5 times in 100.
Statistical Hypothesis Testing
Type I Error
•
•
•
•
9-15
A small a is desirable, other things being
equal.
Chosen in advance, common choices for a are
.10, .05, .025, .01 and .005
(i.e., 10%, 5%, 2.5%, 1% and .5%).
The a risk is the area under the tail(s) of the
sampling distribution.
In a two-sided test, the a risk is split with a/2 in
each tail since there are two ways to reject H0.
Statistical Hypothesis Testing
Type II Error
•
b, the probability of a type II error, is the
probability that the test statistic falls in the
acceptance region even though H0 is false.
b = P(fail to reject H0 | H0 is false)
•
•
9-16
b cannot be chosen in advance because it
depends on a and the sample size.
A small b is desirable, other things being
equal.
Statistical Hypothesis Testing
Power of a Test
•
•
•
The power of a test is the probability that a
false hypothesis will be rejected.
Power = 1 – b
A low b risk means high power.
Power = P(reject H0 | H0 is false) = 1 – b
•
9-17
Larger samples lead to increased power.
Statistical Hypothesis Testing
Relationship Between a and b
•
•
•
•
9-18
Both a small a and a small b are desirable.
For a given type of test and fixed sample
size, there is a trade-off between a and b.
The larger critical value needed to reduce a
risk makes it harder to reject H0, thereby
increasing b risk.
Both a and b can be reduced
simultaneously only by increasing the
sample size.
Statistical Hypothesis Testing
Consequences of a Type II Error
•
Firms are increasingly wary of Type II errror (failing
to recall a product as soon as sample evidence
begins to indicate potential problems.)
Significance versus Importance
•
•
•
9-19
The standard error of most sample estimators
approaches 0 as sample size increases.
In this case, no matter how small, q – q0 will be
significant if the sample size is large enough.
Therefore, expect significant effects even when an
effect is too slight to have any practical
importance.
Testing a Mean:
Known Population Variance
•
•
•
9-20
The hypothesized mean m0 that we are
testing is a benchmark.
The value of m0 does not come from a
sample.
The test statistic compares the sample mean
x with the hypothesized mean m0.
•
The difference between x and m0 is divided by
the standard error of the mean (denoted sx).
•
The test statistic is
Testing a Mean:
Known Population Variance
Testing the Hypothesis
•
•
9-21
Step 1: State the hypotheses
For example, H0: m < 216 mm
H1: m > 216 mm
Step 2: Specify the decision rule
For example, for a = .05
for the right-tail area,
Reject H0 if z > 1.645,
otherwise do not
reject H0
Testing a Mean:
Known Population Variance
Testing the Hypothesis
•
9-22
For a two-tailed test, we split the risk of
Type I error by putting a/2 in each tail.
For example, for a = .05
Testing a Mean:
Known Population Variance
Testing the Hypothesis
9-23
•
Step 3: Calculate the test statistic
•
Step 4: Make the decision
If the test statistic falls in the rejection
region as defined by the critical value, we
reject H0 and conclude H1.
Testing a Mean:
Known Population Variance
Analogy to Confidence Intervals
•
•
9-24
A two-tailed hypothesis test at the 5% level
of significance (a = .05) is exactly
equivalent to asking whether the 95%
confidence interval for the mean includes
the hypothesized mean.
If the confidence interval includes the
hypothesized mean, then we cannot reject
the null hypothesis.
Testing a Mean:
Known Population Variance
Using the p-Value Approach
•
•
•
•
9-25
The p-value is the probability of the sample
result (or one more extreme) assuming that
H0 is true.
The p-value can be obtained using Excel’s
cumulative standard normal function
=NORMSDIST(z)
The p-value can also be obtained from
Appendix C-2.
Using the p-value, we reject H0 if p-value < a.
Testing a Mean:
Unknown Population Variance
Using Student’s t
9-26
•
When the population standard deviation s
is unknown and the population may be
assumed normal, the test statistic follows
the Student’s t distribution with n = n – 1
degrees of freedom.
•
The test statistic is
Testing a Mean:
Unknown Population Variance
Testing a Hypothesis
•
•
9-27
Step 1: State the hypotheses
For example, H0: m = 142
H1: m ≠ 142
Step 2: Specify the decision rule
For example, for a = .10
for a two-tailed area,
Reject H0 if t > 1.714 or
t < -1.714, otherwise
do not reject H0
Testing a Mean:
Unknown Population Variance
Testing a Hypothesis
9-28
•
Step 3: Calculate the test statistic
•
Step 4: Make the decision
If the test statistic falls in the rejection
region as defined by the critical values, we
reject H0 and conclude H1.
Testing a Mean:
Unknown Population Variance
Confidence Intervals versus Hypothesis Test
•
•
9-29
A two-tailed hypothesis test at the 10%
level of significance (a = .10) is equivalent
to a two-sided 90% confidence interval for
the mean.
If the confidence interval does not include
the hypothesized mean, then we reject the
null hypothesis.
Testing a Proportion
•
•
•
•
9-30
To conduct a hypothesis test, we need to know
- the parameter being tested
- the sample statistic
- the sampling distribution of the sample
statistic
The sampling distribution tells us which test
statistic to use.
A sample proportion p estimates the
population proportion p.
Remember that for a large sample, p can be
assumed to follow a normal distribution. If
so, the test statistic is z.
Testing a Proportion
x
number
of
successes
p= =
n
sample size
•
If np0 > 10 and n(1-p0) > 10, then
zcalc =
9-31
p – p0 Where s = p0(1-p0)
p
n
sp
Testing a Proportion
•
•
9-32
The value of p0 that we are testing is a
benchmark such as past experience, an
industry standard, or a product
specification.
The value of p0 does not come from a
sample.
Testing a Proportion
Critical Value
•
•
9-33
The test statistic is compared with a critical
value from a table.
The critical value shows the range of
values for the test statistic that would be
expected by chance if the H0 were true.
Level of Sig. (a)
Two-tailed
test
Right-Tailed
Test
Left-Tailed
Test
.10
+ 1.645
1.282
-1.282
.05
+1.960
1.645
-1.645
.01
+ 2.576
2.326
-2.326
Testing a Proportion
Steps in Testing a Proportion
•
•
Step 1: State the hypotheses
For example, H0: p > .13
H1: p < .13
Step 2: Specify the decision rule
For example, for a = .05
for a left-tail area,
reject H0 if z < -1.645,
otherwise do not
reject H0
Figure 9.12
9-34
Testing a Proportion
Steps in Testing a Proportion
•
For a two-tailed test, we split the risk of
type I error by putting a/2 in each tail.
For example, for a = .05
Figure 9.14
9-35
Testing a Proportion
Steps in Testing a Proportion
•
•
Now, check the normality assumption:
np0 > 10 and n(1-p0) > 10.
Step 3: Calculate the test statistic
p – p0
z=
sp
•
9-36
p0(1-p0)
Where sp =
n
Step 4: Make the decision
If the test statistic falls in the rejection
region as defined by the critical value, we
reject H0 and conclude H1.
Testing a Proportion
Using the p-Value
•
•
•
•
9-37
The p-value is the probability of the sample
result (or one more extreme) assuming that
H0 is true.
The p-value can be obtained using Excel’s
cumulative standard normal function
=NORMSDIST(z)
The p-value can also be obtained from
Appendix C-2.
Using the p-value, we reject H0 if p-value < a.
Testing a Proportion
Using the p-Value
•
•
9-38
The p-value is a direct measure of the level
of significance at which we could reject H0.
Therefore, the smaller the p-value, the more
we want to reject H0.
Testing a Proportion
Calculating a p-Value for a Two-Tailed Test
•
•
For a two-tailed test, we divide the risk into
equal tails. So, to compare the p-value to
a, first combine the p-values in the two tail
areas.
For example, if our test statistic was -1.975,
then
2 x P(z < -1.975) = 2 x .02413 = .04826
At a = .05, we would reject H0 since
p-value = .04826 < a.
9-39
Testing a Proportion
Effect of a
•
No matter which level of significance you use,
the test statistic
remains the
same. For
example, for a
test statistic of
z = 2.152
- 2.576
9-40
- 1.645
1.645
2.576
z = 2.152
Testing a Proportion
Small Samples and Non-Normality
•
In the case where np0 < 10, use MINITAB to
test the hypotheses by finding the exact
binomial probability of a sample proportion p.
For example,
Figure 9.19
9-41
Power Curves and OC Curves
(Optional)
Power Curves for a Mean
•
•
•
Power depends on how far the true value of
the parameter is from the null hypothesis
value.
The further away the true population value is
from the assumed value, the easier it is for
your hypothesis test to detect and the more
power it has.
Remember that
b = P(accept H0 | H0 is false)
Power = P(reject H0 | H0 is false) = 1 – b
9-42
Power Curves and OC Curves
Power Curves for a Mean
•
•
We want power to be as close to 1 as possible.
The values of b and power will vary, depending
on
- the difference between the true mean m and
the
hypothesized mean m0,
- the standard deviation,
- the sample size n and
- the level of significance a
Power = f(m – m0, s, n, a)
9-43
Power Curves and OC Curves
Power Curves for a Mean
Table 9.8
•
•
9-44
We can get more power by increasing a,
but we would then increase the probability
of a type I error.
A better way to increase power is to
increase the sample size n.
Power Curves and OC Curves
Calculating Power
For any given values of m, s, n, and a, and
the assumption that X is normally
distributed, use the following steps to
calculate b and power.
• Step 1: Find the left-tail critical value for
the sample mean.
•
9-45
Power Curves and OC Curves
Calculating Power
9-46
•
The decision rule is:
Reject H0 if x < xcritical
•
The probability of b error is the area to the
right of the critical value xcritical which
represents
P(x > xcritical | m = m0)
Power Curves and OC Curves
Calculating Power
•
Step 2: Express the difference between the
critical value xcritical and the true mean m as
a z-value:
•
Step 3: Find the b risk and power as areas
under the normal curve using Appendix C-2
or Excel. b = P(x > x
|m=m)
critical
0
Power = P(x < xcritical | m = m0) = 1 – b
9-47
Power Curves and OC Curves
Effect of Sample Size
•
9-48
Other things being equal, if sample size
were to increase, b risk would decline and
power would increase because the critical
value xcritical would be closer to the
hypothesized mean m.
Power Curves and OC Curves
Relationship of the Power and OC Curves
•
Here is a family of power curves.
Figure 9.22
9-49
Power Curves and OC Curves
Relationship of the Power and OC Curves
•
9-50
The graph of b risk against this same X-axis
is called the operating characteristic or OC
curve.
Figure 9.23
Power Curves and OC Curves
Power Curve for Tests of a Proportion
•
Power depends on
- the true proportion (p),
- the hypothesized proportion (p0)
- the sample size (n)
- the level of significance (a)
Table 9.10
9-51
Power Curves and OC Curves
Calculating Power
9-52
•
Step 1: Find the left-tail critical value for
the sample proportion.
•
Step 2: Express the difference between the
critical value pcritical and the true proportion
p as a z-value:
Power Curves and OC Curves
Calculating Power
•
Step 3: Find the b risk and power as areas
under the normal curve.
b = P(p < pcritical | p = p0)
Power = P(p > pcritical | m > m0) = 1 – b
9-53
Power Curves and OC Curves
Using LearningStats
•
Create power curves for a mean or
proportion without tedious calculations.
Figure 9.25
9-54
Power Curves and OC Curves
Using Visual Statistics
•
Create power curves for a mean or
proportion without tedious calculations.
Figure 9.26
9-55
Tests for One Variance
•
•
•
Sometimes we want to compare the variance
of a process with a historical benchmark or
other standard.
For a two-tailed test, the hypotheses are:
H0: s2 = s20
H1 = s2 ≠ s20
For a test of one variance, assuming a normal
population, the statistic s2 follows the chisquare distribution with degrees of freedom
equal to n = n – 1.
2
(n – 1)s
=
s2
Reject H0 if c2 < c2lower or if c2 > c2upper
c2
9-56
Applied Statistics in
Business & Economics
End of Chapter 9
9-57
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc.