Hypothesis Testing
Chap 10-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
11.2
Nonstatistical Hypothesis Testing…
In a trial a jury must decide between two hypotheses. n H0: The null hypothesis - The defendant is
innocentn H1: The alternative hypothesis -The defendant
is guiltyThe jury does not know which hypothesis is true. They must make a decision on the basis of evidence presented.
11.3
Research Hypothesis Testing…
The FDA or “science” needs to decide on a new theory, drug, treatment…nH0: The null hypothesis - the current theory, drug, treatment, is as good or betternH1: The alternative hypothesis - the new theory, drug, treatment, should replace the old oneResearchers do not know which hypothesis is true. They must make a decision on the basis of evidence presented.
11.4
Hypothesis Testing Terms
n Convicting the defendant is rejecting the null hypothesis in favor of the alternative hypothesis.
n There is enough evidence to conclude that the defendant is guilty (i.e., there is enough evidence to support the alternative hypothesis).
n If the jury acquits it is stating that there is not enough evidence to support the alternative hypothesis.
n The jury is not saying that the defendant is innocent, only that there is not enough to convict.
n We don’t accept the null hypothesis, we just don’t have sufficient evidence to reject it.
11.5
Hypothesis Testing…
n Two hypotheses, the null and the alternative n Begins with the assumption that the null hypothesis is
true.n Is there enough evidence to infer that the alternative
hypothesis is true, or the null is not likely to be true.n There are two possible decisions:
n Enough evidence to support the alternative hypothesis. Reject the null.
n Not enough evidence to support the alternative hypothesis. Fail to reject the null.
Example: Is the Coin Biased?
n Flip a coin 10 times and get 2 HEAdsn Flip a coin 100 times and get 20 HEAdsn Flip a coin 100 times and get 37 HEAdsn Flip a coin 100 times and get 47 HEAds
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-6
We reject the prior assumption that the coin is unbiased if the number of HEADs is very unlikely under that assumption
Example: Is the Coin Biased?
n H0 : Probability of HEAD = ½n Possible alternatives:
n H1 : Probability of HEAD ≠ ½n H1 : Probability of HEAD is > ½n H1 : Probability of HEAD is < ½
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-7
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-8
Number of HEADs in 100 flipd
μ = 50If H0 is true
If it is unlikely that we get this value ...
... Therefore we reject the
null hypothesis that the coin is
unbiased
Example: Is the Coin Biased?
20
... if in fact the coin was unbiased..
X
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-9
Level of Significance and the Rejection Region
H0: μ ≥ 1/2 H1: μ < 1/2
0
H0: μ ≤ 1/2 H1: μ > 1/2
a
a
Representscritical value
Lower-tail test
Level of significance = a
0Upper-tail test
Two-tail testRejection region is shaded
/2
0
a/2aH0: μ = 1/2 H1: μ ≠ 1/2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-10
What is a Hypothesis?
n A hypothesis is a claim (fact, model, parameter) that can be tested.
n population mean
n population proportion
Example: The mean monthly cell phone bill of this city is μ = $42
Example: The proportion of adults in this city with cell phones is p = .68
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-11
The Null Hypothesis, H0
n States the assumption (numerical) to be testedExample: The average number of TV sets in
U.S. Homes is equal to three ( )
n Is always about a model, not about a sample statistic
3μ:H0 =
3μ:H0 = 3X:H0 =
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Population
Claim: thepopulationmean age is 50.(Null Hypothesis:
REJECT
Supposethe samplemean age is 20: X = 20
SampleNull Hypothesis
20 likely if μ = 50?=Is
Hypothesis Testing Process
If not likely,
Now select a random sample
H0: μ = 50 )
X
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-13
Sampling Distribution of X
μ = 50If H0 is true
If it is unlikely that we would get a sample mean of this value ...
... then we reject the null
hypothesis that μ = 50.
Reason for Rejecting H0
20
... if in fact this werethe population mean…
X
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-14
Level of Significance, a
n Defines the unlikely values of the sample statistic if the null hypothesis is truen Defines rejection region of the sampling
distribution
n Is designated by a , (level of significance)n Typical values are .01, .05, or .10
n Is selected by the researcher at the beginning
n Provides the critical value(s) of the test
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-15
Level of Significance and the Rejection Region
H0: μ ≥ 3 H1: μ < 3
0
H0: μ ≤ 3 H1: μ > 3
a
a
Representscritical value
Lower-tail test
Level of significance = a
0Upper-tail test
Two-tail testRejection region is shaded
/2
0
a/2aH0: μ = 3 H1: μ ≠ 3
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-16
Errors in Making Decisions
n Type I Errorn Reject a true null hypothesisn Considered a serious type of error
The probability of Type I Error is an Called level of significance of the testn Set by researcher in advance
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-17
Errors in Making Decisions
n Type II Errorn Fail to reject a false null hypothesis
The probability of Type II Error is β
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-18
Outcomes and Probabilities
Actual SituationDecisionDo NotReject
H0
No error(1 - )a
Type II Error( β )
RejectH0
Type I Error( )a
Possible Hypothesis Test Outcomes
H0 FalseH0 True
Key:Outcome
(Probability) No Error( 1 - β )
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-19
Type I & II Error Relationship
§ Type I and Type II errors can not happen atthe same time§ Type I error can only occur if H0 is true
§ Type II error can only occur if H0 is false
If Type I error probability ( a ) , then
Type II error probability ( β )
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-20
Factors Affecting Type II Error
n All else equal,n β when the difference between
hypothesized parameter and its true value
n β when a
n β when σ
n β when n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-21
Power of the Test
n The power of a test is the probability of rejecting a null hypothesis that is false
n i.e., Power = P(Reject H0 | H1 is true)
n Power of the test increases as the sample size increases
Select your test
n Testing is a bit like finding the right recipe based on these ingredients: n Questionn Data typen Sample sizen Variance known? Variance of several groups equal?
n Good news: Plenty of tables available, e.g.,n http://www.ats.ucla.edu/stat/mult_pkg/whatstat/de
fault.htm (with examples in R, SAS, Stata, SPSS)n http://sites.stat.psu.edu/~ajw13/stat500_su_res/not
es/lesson14/images/summary_table.pdf
Example of a table of tests
http://sites.stat.psu.edu/~ajw13/stat500_su_res/notes/lesson14/images/summary_table.pdf
A common test: One-sample t-test
n When: Estimating a mean, comparing mean to a hypothetical value
n Requirements: Data approx. normal or sample size > 30
n Setup:
n Evaluation: Compare t-statistic (table, excel) to your value and accept / reject null hypothesis
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-26
Hypothesis Tests for the Mean
s Known s Unknown
Hypothesis Tests for µ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-27
Test of Hypothesisfor the Mean (σ Known)
n Convert sample result ( ) to a z value
The decision rule is:
α0
0 z
nσμxz if H Reject >
-=
σ Known σ Unknown
Hypothesis Tests for µ
Consider the test
00 μμ:H =
01 μμ:H >
(Assume the population is normal)
x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-28
Reject H0Do not reject H0
Decision Rule
a
zα0
μ0
H0: μ = μ0
H1: μ > μ0
Critical value
Z
α0
0 z
nσμxz if H Reject >
-=
nσ/ZμX if H Reject α00 +>
nσzμ α0 +
Alternate rule:
x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-29
p-Value Approach to Testing
n p-value: Probability of obtaining a test statistic more extreme ( ≤ or ³ ) than the observed sample value given H0 is true
n Also called observed level of significance
n Smallest value of a for which H0 can be rejected
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-30
p-Value Approach to Testing
n Convert sample result (e.g., ) to test statistic (e.g., z statistic )
n Obtain the p-valuen For an upper
tail test:
n Decision rule: compare the p-value to a
n If p-value < a , reject H0
n If p-value ³ a , do not reject H0
(continued)x
)μμ | nσ/μ-x P(Z
true) is H that given , nσ/μ-x P(Z value-p
00
00
=>=
>=
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-31
Example: Upper-Tail Z Test for Mean (s Known)
A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume s = 10 is known)
H0: μ ≤ 52 the average is not over $52 per monthH1: μ > 52 the average is greater than $52 per month
(i.e., sufficient evidence exists to support the manager’s claim)
Form hypothesis test:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-32
Reject H0Do not reject H0
n Suppose that a = .10 is chosen for this test
Find the rejection region:
a = .10
1.280
Reject H0
Example: Find Rejection Region(continued)
1.28nσ/μxz if H Reject 0
0 >-
=
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-33
Obtain sample and compute the test statistic
Suppose a sample is taken with the following results: n = 64, x = 53.1 (s=10 was assumed known)
n Using the sample results,
0.88
6410
5253.1
nσμxz 0 =
-=
-=
Example: Sample Results(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-34
Reject H0Do not reject H0
Example: Decision
a = .10
1.280
Reject H0
Do not reject H0 since z = 0.88 < 1.28i.e.: there is not sufficient evidence that the
mean bill is over $52
z = 0.88
Reach a decision and interpret the result:(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-35
Reject H0
a = .10
Do not reject H0 1.280
Reject H0
Z = .88
Calculate the p-value and compare to a(assuming that μ = 52.0)
(continued)
.1894
.810610.88)P(z
6410/52.053.1zP
52.0) μ | 53.1xP(
=
-=³=
÷ø
öçè
æ -³=
=³p-value = .1894
Example: p-Value Solution
Do not reject H0 since p-value = .1894 > a = .10
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-36
One-Tail Tests
n In many cases, the alternative hypothesis focuses on one particular direction
H0: μ ≥ 3 H1: μ < 3
H0: μ ≤ 3 H1: μ > 3
This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3
This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-37
Reject H0Do not reject H0
Upper-Tail Tests
a
zα0μ
H0: μ ≤ 3 H1: μ > 3
n There is only one critical value, since the rejection area is in only one tail
Critical value
Zx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-38
Reject H0 Do not reject H0
n There is only one critical value, since the rejection area is in only one tail
Lower-Tail Tests
a
-za 0
μ
H0: μ ≥ 3 H1: μ < 3
Z
Critical value
x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-39
Do not reject H0 Reject H0Reject H0
n There are two critical values, defining the two regions of rejection
Two-Tail Tests
a/2
0
H0: μ = 3 H1: μ ¹ 3
a/2
Lower critical value
Uppercritical value
3
z
x
-za/2 +za/2
n In some settings, the alternative hypothesis does not specify a unique direction
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-40
Hypothesis Testing ExampleTest the claim that the true mean # of TV
sets in US homes is equal to 3.(Assume σ = 0.8)
n State the appropriate null and alternativehypothesesn H0: μ = 3 , H1: μ ≠ 3 (This is a two tailed test)
n Specify the desired level of significancen Suppose that a = .05 is chosen for this test
n Choose a sample sizen Suppose a sample of size n = 100 is selected
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-41
2.0.08.16
1000.8
32.84
nσμXz 0 -=
-=
-=
-=
Hypothesis Testing Example
n Determine the appropriate techniquen σ is known so this is a z test
n Set up the critical valuesn For a = .05 the critical z values are ±1.96
n Collect the data and compute the test statisticn Suppose the sample results are
n = 100, x = 2.84 (σ = 0.8 is assumed known)So the test statistic is:
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-42
Reject H0 Do not reject H0
n Is the test statistic in the rejection region?
a = .05/2
-z = -1.96 0
Reject H0 if z < -1.96 or z > 1.96; otherwise do not reject H0
Hypothesis Testing Example(continued)
a = .05/2
Reject H0
+z = +1.96
Here, z = -2.0 < -1.96, so the test statistic is in the rejection region
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-43
n Reach a decision and interpret the result
-2.0
Since z = -2.0 < -1.96, we reject the null hypothesisand conclude that there is sufficient evidence that the mean number of TVs in US homes is not equal to 3
Hypothesis Testing Example(continued)
Reject H0 Do not reject H0
a = .05/2
-z = -1.96 0
a = .05/2
Reject H0
+z = +1.96
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-44
.0228
a/2 = .025
Example: p-Value
n Example: How likely is it to see a sample mean of 2.84 (or something further from the mean, in either direction) if the true mean is µ = 3.0?
-1.96 0-2.0
.02282.0)P(z
.02282.0)P(z
=>
=-<
Z1.962.0
x = 2.84 is translated to a z score of z = -2.0
p-value
= .0228 + .0228 = .0456
.0228
a/2 = .025
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-45
n Compare the p-value with an If p-value < a , reject H0
n If p-value ³ a , do not reject H0
Here: p-value = .0456a = .05
Since .0456 < .05, we reject the null hypothesis
(continued)
Example: p-Value
.0228
a/2 = .025
-1.96 0-2.0
Z1.962.0
.0228
a/2 = .025
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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-52
t Test of Hypothesis for the Mean (σ Unknown)
n Convert sample result ( ) to a t test statistic
σ Known σ Unknown
Hypothesis Tests for µ
x
The decision rule is:
α , 1-n0
0 t
nsμxt if H Reject >
-=
Consider the test
00 μμ:H =
01 μμ:H >
(Assume the population is normal)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-53
t Test of Hypothesis for the Mean (σ Unknown)
n For a two-tailed test:
The decision rule is:
α/2 , 1-n0
α/2 , 1-n0
0 t
nsμxt if or t
nsμxt if H Reject >
-=-<
-=
Consider the test
00 μμ:H =
01 μμ:H ¹
(Assume the population is normal, and the population variance is unknown)
(continued)
What is a T-distribution?
n A t-distribution is like a Z distribution, except has slightly fatter tails to reflect the uncertainty added by estimating s.
n The bigger the sample size (i.e., the bigger the sample size used to estimate s), then the closer t becomes to Z.
n If n>100, t approaches Z.
The T probability density function
Where:v is the degrees of freedom
(gamma) is the Gamma function is the constant Pi (3.14...)
Student’s t Distribution
t0
t (df = 5)
t (df = 13)t-distributions are bell-
shaped and symmetric, but have ‘fatter’ tails than the
normal
Standard Normal
(t with df = ¥)
Note: t Z as n increases
from “Statistics for Managers” Using Microsoft® Excel 4th Edition, Prentice-Hall 2004
Student’s t Table
Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
t0 2.920The body of the table contains t values, not
probabilities
Let: n = 3 df = n - 1 = 2
a = .10a/2 =.05
a/2 = .05
from “Statistics for Managers” Using Microsoft® Excel 4th Edition, Prentice-Hall 2004
t distribution values
With comparison to the Z valueConfidence t t t ZLevel (10 d.f.) (20 d.f.) (30 d.f.) ____
.80 1.372 1.325 1.310 1.28
.90 1.812 1.725 1.697 1.64
.95 2.228 2.086 2.042 1.96
.99 3.169 2.845 2.750 2.58
Note: t Z as n increases
from “Statistics for Managers” Using Microsoft® Excel 4th Edition, Prentice-Hall 2004
Summary: Single population mean (small n, normality)
n Hypothesis test:
n Confidence Intervalnstx
nmean nullmean observed
1-
=-
)(* tmean observed interval confidence /21,-n nsx
a±=
Summary: Single population mean (large n)
n Hypothesis test:
n Confidence IntervalnstZx
nmean nullmean observed
1-
=@ -
)(*]Z t[mean observed interval confidence /2/21,-n nsx
aa @±=
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-61
Example: Two-Tail Test(s Unknown)
The average cost of a hotel room in New York is said to be $168 per night. A random sample of 25 hotels resulted in x = $172.50 and s = $15.40. Test at the a = 0.05 level.(Assume the population distribution is normal)
H0: μ = 168 H1: μ ¹ 168
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-62
n a = 0.05
n n = 25n s is unknown, so
use a t statisticn Critical Value:
t24 , .025 = ± 2.0639
Example Solution: Two-Tail Test
Do not reject H0: not sufficient evidence that true mean cost is different than $168
Reject H0Reject H0
a/2=.025
-t n-1,α/2Do not reject H0
0
a/2=.025
-2.0639 2.0639
1.46
2515.40
168172.50
nsμxt 1n =
-=
-=-
1.46
H0: μ = 168 H1: μ ¹ 168
t n-1,α/2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-63
Tests of the Population Proportion
n Involves categorical variables
n Two possible outcomesn “Success” (a certain characteristic is present)
n “Failure” (the characteristic is not present)
n Fraction or proportion of the population in the “success” category is denoted by P
n Assume sample size is large
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-64
Proportions
n Sample proportion in the success category is denoted by
n
n When nP(1 – P) > 9, can be approximated by a normal distribution with mean and standard deviationn
sizesamplesampleinsuccessesofnumber
nxp ==ˆ
Pμ =p̂nP)P(1σ -
=p̂
(continued)
p̂
p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-65
n The sampling distribution of is approximately normal, so the test statistic is a z value:
Hypothesis Tests for Proportions
n)P(1P
Ppz00
0
--
=ˆ
nP(1 – P) > 9
Hypothesis Tests for P
Exact test
p̂
nP(1 – P) < 9
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-66
Example: Z Test for Proportion
A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the a = .05 significance level.
Check: Our approximation for P is
= 25/500 = .05
nP(1 - P) = (500)(.05)(.95)= 23.75 > 9
p̂
ü
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-67
Z Test for Proportion: Solution
a = .05 n = 500, = .05
Reject H0 at a = .05
H0: P = .08 H1: P ¹ .08
Critical Values: ±1.96
Test Statistic:
Decision:
Conclusion:
z0
Reject Reject
.025.025
1.96-2.47
There is sufficient evidence to reject the company’s claim of 8% response rate.
2.47
500.08).08(1.08.05
n)P(1P
Ppz00
0 -=--
=-
-=
ˆ
-1.96
p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-68
Do not reject H0 Reject H0Reject H0
a/2 = .025
1.960
Z = -2.47
Calculate the p-value and compare to a(For a two sided test the p-value is always two sided)
(continued)
0.01362(.0068)
2.47)P(Z2.47)P(Z
==
³+-£
p-value = .0136:
p-Value Solution
Reject H0 since p-value = .0136 < a = .05
Z = 2.47
-1.96
a/2 = .025
.0068.0068
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-69
n Recall the possible hypothesis test outcomes:Actual Situation
Decision
Do Not Reject H0
No error(1 - )a
Type II Error( β )
Reject H0Type I Error
( )a
H0 FalseH0 TrueKey:
Outcome(Probability)
No Error( 1 - β )
n β denotes the probability of Type II Error n 1 – β is defined as the power of the test
Power = 1 – β = the probability that a false null hypothesis is rejected
Power of the Test
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-70
Type II Error
or
The decision rule is:
α0
0 zn/σμxz if H Reject >
-=
00 μμ:H =
01 μμ:H >
Assume the population is normal and the population variance is known. Consider the test
nσ/Zμxx if H Reject α0c0 +>=
If the null hypothesis is false and the true mean is μ*, then the probability of type II error is
÷÷ø
öççè
æ -<==<=
n/σ*μxzPμ*)μ|xxP(β c
c
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-71
Reject H0: μ ³ 52
Do not reject H0 : μ ³ 52
Type II Error Examplen Type II error is the probability of failing
to reject a false H0
5250
Suppose we fail to reject H0: μ ³ 52when in fact the true mean is μ* = 50
a
cx!!
cx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-72
Reject H0: μ ³ 52
Do not reject H0 : μ ³ 52
Type II Error Example
n Suppose we do not reject H0: μ ³ 52 when in fact the true mean is μ* = 50
5250
This is the true distribution of x if μ = 50
This is the range of x where H0 is not rejected
(continued)
cx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-73
Reject H0: μ ³ 52
Do not reject H0 : μ ³ 52
Type II Error Example
n Suppose we do not reject H0: μ ³ 52 when in fact the true mean is μ* = 50
a
5250
β
Here, β = P( x ³ ) if μ* = 50
(continued)
cx
cx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-74
Reject H0: μ ³ 52
Do not reject H0 : μ ³ 52
n Suppose n = 64 , σ = 6 , and a = .05
a
5250
So β = P( x ³ 50.766 ) if μ* = 50
Calculating β
50.7666461.64552
nσzμx α0c =-=-=
(for H0 : μ ³ 52)
50.766
cx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-75
Reject H0: μ ³ 52
Do not reject H0 : μ ³ 52
.1539.3461.51.02)P(z64
65050.766zP50)μ*|50.766xP( =-=³=
÷÷÷
ø
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³==³
n Suppose n = 64 , σ = 6 , and a = .05
a
5250
Calculating β(continued)
Probability of type II error:
β = .1539
cx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-76
If the true mean is μ* = 50,n The probability of Type II Error = β = 0.1539n The power of the test = 1 – β = 1 – 0.1539 = 0.8461
Power of the Test Example
Actual SituationDecision
Do Not Reject H0
No error1 - a = 0.95
Type II Errorβ = 0.1539
Reject H0Type I Errora = 0.05
H0 FalseH0 TrueKey:
Outcome(Probability)
No Error1 - β = 0.8461
(The value of β and the power will be different for each μ*)