type ii error in hypo test.pdf
Post on 04-Jun-2018
231 Views
Preview:
TRANSCRIPT
-
8/14/2019 type II Error in hypo test.pdf
1/15
1
1Slide
Chapter 9, Part BHypothesis Tests
2Slide
Learning objectives
1. Able to do hypothesis test about Population Proportion
2. Calculatethe Probability of Type II Errors
3. Understand power of the test
4. Determinethe Sample Size for Hypothesis Tests About aPopulation Mean
-
8/14/2019 type II Error in hypo test.pdf
2/15
2
3Slide
n The equality part of the hypotheses always appears
in the null hypothesis.
n In general, a hypothesis test about the value of a
population proportionp must take one of the
following three forms (wherep0 is the hypothesized
value of the population proportion).
A Summary of Forms for Null and AlternativeHypotheses About a Population Proportion
One-tailed(lower tail)
One-tailed(upper tail)
Two-tailed
0 0:H p p0 0:H p p
0:aH p p> 0:aH p p>0:aH p p< 0:aH p p 5 and n(1 p) > 5
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
3/15
3
5Slide
n Rejection Rule: p Value Approach
H0: pz
Reject H0 if z < -z
Reject H0 if z < -z/2 or z >z/2
H0: p>p0
H0: p=p0
Tests About a Population Proportion
Reject H0 ifp value <
n Rejection Rule: Critical Value Approach
L.O.Test for pPowerSample size
6Slide
n Example: National Safety Council
For a Christmas and 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.
Two-Tailed Test About aPopulation Proportion
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
4/15
4
7Slide
A sample of 120 accidents showed that
67 were caused by drunk driving. Use
these data to test the NSC!s claim with
= .05.
Two-Tailed Test About aPopulation Proportion
n Example: National Safety Council
L.O.Test for pPowerSample size
8Slide
Two-Tailed Test About aPopulation Proportion
1. Determine the hypotheses.
2. Specify the level of significance.
3. Compute the value of the test statistic.
= .05
n p Value and Critical Value Approaches
0 : .5H p =0 : .5H p =: .5aH p : .5aH p
= = =0
(67/120) .5 1.28
.045644p
p pz
= = =0
(67/120) .5 1.28
.045644p
p pz
0 0(1 ) .5(1 .5)
.045644120pp p
n
= = =
0 0(1 ) .5(1 .5)
.045644120p
p p
n
= = =a commonerror is using
in thisformulapp
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
5/15
5
9Slide
n pValue Approach
4. Compute thep -value.
5. Determine whether to reject H0.
Becausepvalue = .2006 > = .05, we cannot reject H0.
Two-Tailed Test About aPopulation Proportion
For z = 1.28, cumulative probability = .8997
pvalue = 2(1 .8997) = .2006
L.O.Test for pPowerSample size
10Slide
Two-Tailed Test About aPopulation Proportion
n Critical Value Approach
5. Determine whether to reject H0.
For /2 = .05/2 = .025, z.025 = 1.96
4. Determine the critical value and rejection rule.
Reject H0 if z < -1.96 or z > 1.96
Because 1.278 > -1.96 and < 1.96, we cannot reject H0.
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
6/15
6
11Slide
In-class exercise
n #35 (p.368) (two-tail test)
n #36 (p.368) (one-tail test)
L.O.Test for pPowerSample size
12Slide
Statistical power Type II errorL.O.Test for pPowerSample size
n As said earlier (Part a.), the conclusion that theresearch hypothesis is truecomes from the test thatcontradict (or reject) the null hypothesis.
n When a test fails to reject a null hypothesis (i.e.,acceptsH
0), type II error () needs to be checked.
n A Type II error is acceptingH0 when it is false.
n When a test rejectH0, possible error is Type I error
that can be controlled easily.
n Article by Sawyer and Ball (1981)
-
8/14/2019 type II Error in hypo test.pdf
7/15
7
13Slide
Calculating the Probability of a Type II Errorin Hypothesis Tests About a Population Mean
1. Formulate the null and alternative hypotheses.
3. Using the rejection rule, solve for the value of thesample mean corresponding to the critical value ofthe test statistic.
2. Using the critical value approach, use the level ofsignificance to determine the critical value andthe rejection rule for the test.
L.O.Test for pPowerSample size
14Slide
Calculating the Probability of a Type II Errorin Hypothesis Tests About a Population Mean
4. Use the results from step 3 to state the values of the
sample mean that lead to the acceptance of H0; thisdefines the acceptance region.
5. Using the sampling distribution of for a value of
satisfying the alternative hypothesis, and the acceptance
region from step 4, compute the probability that the
sample mean will be in the acceptance region. (This is
the probability of making a Type II error at the chosen
level of .)
xx
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
8/15
8
15Slide
n Example: Metro EMS (revisited)
The EMS director wants toperform a hypothesis test, with a.05 level of significance, to determine
whether or not the service goal of 12 minutes or less isbeing achieved.
Recall that the response times fora random sample of 40 medicalemergencies were tabulated. Thesample mean is 13.25 minutes.The population standard deviationis believed to be 3.2 minutes.
Calculating the Probabilityof a Type II Error
L.O.Test for pPowerSample size
16Slide
=
121.645
3.2 / 40
xz
=
121.645
3.2 / 40
xz
+ =
3.212 1.645 12.8323
40x
+ =
3.212 1.645 12.8323
40x
4. We will accept H0 when x < 12.8323
3. Value of the sample mean that identifiesthe rejection region:
2. Rejection rule is: Reject H0 if z > 1.645
1. Hypotheses are: H0: < 12 and Ha: > 12
Calculating the Probabilityof a Type II Error
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
9/15
9
17Slide
12.0001 1.645 .9500 .050012.4 0.85 .8023 .1977
12.8 0.06 .5239 .476112.8323 0.00 .5000 .500013.2 -0.73 .2327 .767313.6 -1.52 .0643 .935714.0 -2.31 .0104 .9896
12.8323
3.2 / 40z
=
12.8323
3.2 / 40z
=
Values of 1-
5. Probabilities that the sample mean will bein the acceptance region:
Calculating the Probabilityof a Type II Error
L.O.Test for pPowerSample size
18Slide
Calculating the Probabilityof a Type II Error
n Calculating the Probability of a Type II Error
l When the true population mean is far abovethe null hypothesis value of 12, there is a low
probability that we will make a Type II error.
l When the true population mean is close tothe null hypothesis value of 12, there is a highprobability that we will make a Type II error.
Observations about the preceding table:
Example: = 12.0001, = .9500
Example: = 14.0, = .0104
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
10/15
10
19Slide
Power of the Test
n The probability of correctly rejecting H0 when it isfalse is called the power of the test.
n For any particular value of , the power is 1 .
n We can show graphically the power associatedwith each value of ; such a graph is called apower curve. (See next slide.)
L.O.Test for pPowerSample size
20Slide
Power Curve
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
11.5 12.0 12.5 13.0 13.5 14.0 14.5
ProbabilityofCorrectly
RejectingNullHypothesis
H0 False
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
11/15
11
21Slide
In-class exercise
n #46 (p.374)
n #50 (p.375)
L.O.Test for pPowerSample size
22Slide
n The specified level of significance determines the
probability of making a Type I error.
n By controlling the sample size, the probability ofmaking a Type II error is controlled.
Determining the Sample Size fora Hypothesis Test About a Population Mean
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
12/15
12
23Slide
0
a
xx
xx
Samplingdistributionof whenH0 is trueand = 0
xx
Samplingdistributionof when
H0 is falseand a > 0
xx
Reject H0
c
c
H0: < 0Ha: > 0
=
x n
=
x nNote:
Determining the Sample Size fora Hypothesis Test About a Population Mean
L.O.Test for pPowerSample size
24Slide
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 H0a = value of the population mean used for the
Type II error
nz z
a
=+
( )
( )
2 2
02
nz z
a
=+
( )
( )
2 2
02
Determining the Sample Size fora Hypothesis Test About a Population Mean
Note: In a two-tailed hypothesis test, use z /2 not z
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
13/15
13
25Slide
Relationship Among , , and n
n Once two of the three values are known, the othercan be computed.
n For a given level of significance , increasing thesample size n will reduce .
n For a given sample size n, decreasing will increase, whereas increasing will decrease b.
L.O.Test for pPowerSample size
26Slide
Determining the Sample Size fora Hypothesis Test About a Population Mean
n 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 amwilling to risk an = .05 probability of rejecting H0.
"If the mean response time is 0.75 minutes over thespecification ( = 12.75), I am willing to risk a = .10probability of not rejecting H0.
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
14/15
14
27Slide
Determining the Sample Size fora Hypothesis Test About a Population Mean
= .05, = .10
z = 1.645, z = 1.28
0 = 12, a = 12.75
= 3.2
2 2 2 2
2 2
0
( ) (1.645 1.28) (3.2)155.75 156
( ) (12 12.75)a
z zn
+ += = =
2 2 2 2
2 2
0
( ) (1.645 1.28) (3.2)155.75 156
( ) (12 12.75)a
z zn
+ += = =
L.O.Test for pPowerSample size
28Slide
In-class exercise
n #54 (p.379)
n #55 (p.379)
L.O.Test for pPowerSample size
-
8/14/2019 type II Error in hypo test.pdf
15/15
15
29Slide
End of Chapter 9, Part B
top related