Unformatted text preview: Hypothesis Tests II
AS&W – Chapter 9 Population Proportion Hypothesis Testing and Decision Making Calculating the Probability of Type II Errors Determining the Sample Size for
Hypothesis Tests About a Population Mean 1 A Summary of Forms for Null and Alternative
Hypotheses About a Population Proportion The equality part of the hypotheses always appears
in the null hypothesis. In general, a hypothesis test about the value of a
population proportion p must take one of the
following three forms (where p0 is the hypothesized
value of the population proportion). H 0: p p0
H a: p p0 H 0: p p0
H a: p p0 H 0: p p0
H a: p p0 Onetailed Onetailed
(lower tail) (upper tail) Twotailed 2 Tests About a Population Proportion Test Statistic p p0
z
p
where: p0 (1 p0 )
p n
assuming np > 5 and n(1 – p) > 5 3 Tests About a Population Proportion Rejection Rule: p –Value Approach
Reject H0 if p –value < Rejection Rule: Critical Value Approach
H0: pp0 Reject H0 if z > z
H0: pp Reject H0 if z < z
H0: pp Reject H0 if z < z2 or z > z 4 TwoTailed Test About a
Population Proportion Example: National Safety Council
For New Year’s week, the
National Safety Council estimated that
500 people would be killed and 25,000
injured on the nation’s roads. The
NSC claimed that 50% of the
accidents would be caused by
drunk driving. 5 TwoTailed Test About a
Population Proportion Example: National Safety Council
A sample of 120 accidents showed that
67 were caused by drunk driving. Use
these data to test the NSC’s claim with = .05. 6 TwoTailed Test About a
Population Proportion p –Value and Critical Value Approaches H 0: p .5
1. Determine the hypotheses.
H a: p .5 = .05
2. Specify the level of significance. 3. Compute the value of the test statistic. a common
error is
pusing
in this
formula p0(1 p0 )
.5(1 .5)
p .045644
n
120
p p0 (67/ 120) .5
z 1.28
p
.045644
7 TwoTailed Test About a
Population Proportion pValue Approach
4. Compute the p value.
For z = 1.28, cumulative probability = .8997
p–value = 2(1 .8997) = .2006
5. Determine whether to reject H0. Because p–value = .2006 > = .05, we cannot reje 8 TwoTailed Test About a
Population Proportion Critical Value Approach
4. Determine the criticals value and
rejection rule.
For /2 = .05/2 = .025, z.025 = 1.96
Reject H0 if z < 1.96 or z > 1.96
5. Determine whether to reject H0. Because 1.278 > 1.96 and < 1.96, we cannot rejec 9 Hypothesis Testing and Decision Making In many decisionmaking situations the decision
maker may want, and in some cases may be forced,
to take action with both the conclusion do not reject
H0 and the conclusion reject H0. In such situations, it is recommended that the
hypothesistesting procedure be extended to
include
consideration of making a Type II error. 10 Calculating the Probability of a Type II
Error
in Hypothesis Tests About a Population
1. Formulate the null Mean
and alternative hypotheses.
2. Using the critical value approach, use the level of
significance to determine the critical value and
the rejection rule for the test.
3. Using the rejection rule, solve for the value of the
sample mean corresponding to the critical value of
the test statistic. 11 Calculating the Probability of a Type II
Error
in Hypothesis Tests About a Population
4. Use the results fromMean
step 3 to state the values of the
sample mean that lead to the acceptance of H0; this
defines the acceptance region. 5. Using the sampling distributionx of
for a value of satisfying the alternative hypothesis, and the accep
region from step 4, compute the probability that the
sample mean will be in the acceptance region. (Thi
the probability of making a Type II error at the chose
level of .) 12 Calculating the Probability
of a Type II Error Example: Metro EMS (revisited) Recall that the response times for
a random sample of 40 medical
emergencies were tabulated. The
sample mean is 13.25 minutes.
The population standard deviation
is believed to be 3.2 minutes.
The EMS director wants to
perform a hypothesis test, with a
.05 level of significance, to determine
whether or not the service goal of 12 minutes or less is
being achieved.
13 Calculating the Probability
of a Type II Error
1. Hypotheses are: H0: 12 and
H rule is: Reject H0 if z > 1.645
2.a:
Rejection
3. Value of the sample mean that identifies
the rejection region:
x
z
3.2/ 12
1.645
40 3.2 x 12 1.645 12.8323 40 4. We will accept H0 when x < 12.8323 14 Calculating the Probability
of a Type II Error
5. Probabilities that the sample mean will be
in the acceptance region: 12.8323 z
Values of 3.2/ 40
14.0
13.6
13.2
12.8323
12.8
12.4
12.0001 2.31
1.52
0.73
0.00
0.06
0.85
1.645 .0104
.0643
.2327
.5000
.5239
.8023
.9500 .9896
.9357
.7673
.5000
.4761
.1977
.0500 15 Calculating the Probability
of a Type II Error Calculating the Probability of a Type II Error
Observations about the preceding table: When the true population mean is close to
the null hypothesis value of 12, there is a high
probability that we will make a Type II error.
Example: = 12.0001, = .9500 When the true population mean is far
above
the null hypothesis value of 12, there is a
low
Example: = 14.0, = .0104
probability that we will make a Type II
error. 16 Power of the Test The probability of correctly rejecting H0 when it is
false is called the power of the test. For any particular value of , the power is 1 – . We can show graphically the power associated
with each value of ; such a graph is called a
power curve. (See next slide.) 17 Power Curve
Probability of Correctly
Rejecting Null Hypothesis 1.00
0.90
0.80 H0 False 0.70
0.60
0.50
0.40
0.30
0.20
0.10 0.00
11.5 12.0 12.5 13.0 13.5 14.0 14.5 18 Determining the Sample Size for a
Hypothesis Test About a Population Mean The specified level of significance determines the
probability of making a Type I error. By controlling the sample size, the probability of
making a Type II error is controlled. 19 Determining the Sample Size for a
Hypothesis Test About a Population Mean
Sampling
distribution
of x when
H0 is true
and = 0 c
Reject H0 x 0
Note: x H0: 0
Ha: n Sampling
distribution
of x when
H0 is false
and a > 0 c a x
20 Determining the Sample Size for a
Hypothesis Test About a Population Mean
n ( z z ) 2 2
( 0 a )2 where
z = z value providing an area of in the tail
z = z value providing an area of in the tail
= population standard deviation
0 = value of the population mean in H0
a = value of the population mean used for the
Type II error Note: In a twotailed hypothesis test, use z /2 not z
21 Relationship Among , , and n Once two of the three values are known, the
other can be computed.
For a given level of significance , increasing
the sample size n will reduce .
For a given sample size n, decreasing will
increase , whereas increasing will decrease
. 22 Determining the Sample Size for a
Hypothesis Test About a Population Mean Let’s assume that the director of medical
services makes the following statements about the
allowable probabilities for the Type I and Type II
errors: •If the mean response time is = 12 minutes, I
am willing to risk an = .05 probability of
rejecting
H0. response time is 0.75 minutes over
•
If the mean
the specification ( = 12.75), I am willing to risk
a = .10 probability of not rejecting H0. 23 Determining the Sample Size for a
Hypothesis Test About a Population Mean = .05, = .10
z = 1.645, z = 1.28
0 = 12, a = 12.75
= 3.2 ( z z )2 2 (1.645 1.28) 2 (3.2) 2
n 155.75 156
2
2
( 0 a )
(12 12.75) 24 ...
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 Spring '13
 aslan
 Null hypothesis, Statistical hypothesis testing, H0, Type I and type II errors, type ii error

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