Lesson 5- Lesson 5-1 Statistics for Management Lesson 5 Fundamentals of Hypothesis Testing
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Lesson 5-Lesson 5-11
Statistics for Management
Lesson 5
Fundamentals
of Hypothesis Testing
Lesson. 5 - 2
Lesson Topics1. What is a Hypothesis?
Hypothesis Testing Methodology
Hypothesis Testing Process
Level of Significance, a; Errors in Making Decisions
2. Hypothesis Testing: Steps
3. Hypothesis Testing for the Mean
Connection to Confidence Interval Estimation Hypothesis Testing Methodology
4. Hypothesis Testing for the Proportion
Lesson. 5 - 3
A hypothesis is an assumption about the population parameter.
A parameter is a Population mean or proportion
The parameter must be identified before analysis.
I assume the mean GPA of this class is 3.5!
© 1984-1994 T/Maker Co.
1. What is a Hypothesis?
Lesson. 5 - 4
• States the Assumption (numerical) to be tested
e.g. The average # TV sets in US homes is at least 3 (H0: 3)
• Begin with the assumption that the null hypothesis is TRUE.
(Similar to the notion of innocent until proven guilty)
The Null Hypothesis, H0
•Always contains the ‘ = ‘ sign
•The Null Hypothesis may or may not be rejected.
Lesson. 5 - 5
• Is the opposite of the null hypothesise.g. The average # TV sets in US
homes is less than 3 (H1: < 3)
• Never contains the ‘=‘ sign
• The Alternative Hypothesis may or may not be accepted
The Alternative Hypothesis, H1
Lesson. 5 - 6
Steps: State the Null Hypothesis (H0: 3) State its opposite, the Alternative
Hypothesis (H1: < 3)Hypotheses are mutually exclusive &
exhaustiveSometimes it is easier to form the
alternative hypothesis first.
Identify the Problem
Lesson. 5 - 7
Population
Assume thepopulationmean age is 50.(Null Hypothesis)
REJECT
The SampleMean Is 20
SampleNull Hypothesis
50?20 XIs
Hypothesis Testing Process
No, not likely!
Lesson. 5 - 8
Sample Mean = 50
Sampling DistributionIt is unlikely that we would get a sample mean of this value ...
... if in fact this were the population mean.
... Therefore, we reject the null
hypothesis that = 50.
20H0
Reason for Rejecting H0
Lesson. 5 - 9
• Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling
Distribution
• Designated (alpha) Typical values are 0.01, 0.05, 0.10
• Selected by the Researcher at the Start
• Provides the Critical Value(s) of the Test
Level of Significance,
Lesson. 5 - 10
Level of Significance, and the Rejection Region
H0: 3
H1: < 30
0
0
H0: 3
H1: > 3
H0: 3
H1: 3
/2
Critical Value(s)
Rejection Regions
Lesson. 5 - 11
• Type I Error Reject True Null Hypothesis Has Serious Consequences Probability of Type I Error Is
Called Level of Significance
• Type II Error Do Not Reject False Null Hypothesis Probability of Type II Error Is (Beta)
Errors in Making Decisions
Lesson. 5 - 12
H0: Innocent
Jury Trial Hypothesis Test
Actual Situation Actual Situation
Verdict Innocent Guilty Decision H0 True H0 False
Innocent Correct ErrorDo NotReject
H0
1 - Type IIError ( )
Guilty Error Correct RejectH0
Type IError( )
Power(1 - )
Result Possibilities
Lesson. 5 - 13
Reduce probability of one error and the other one goes up.
& Have an Inverse Relationship
Lesson. 5 - 14
• True Value of Population Parameter Increases When Difference Between Hypothesized
Parameter & True Value Decreases
• Significance Level Increases When Decreases
• Population Standard Deviation Increases When Increases
• Sample Size n Increases When n Decreases
Factors Affecting Type II Error,
n
Lesson. 5 - 15
1. State H0 H0 : 3
2. State H1 H1 :
3. Choose = .05
4. Choose n n = 100
5. Choose Test: Z Test (or p Value)
2. Hypothesis Testing: Steps
Test the Assumption that the true mean # of TV sets in US homes is at least 3.
Lesson. 5 - 16
6. Set Up Critical Value(s) Z = -1.645
7. Collect Data 100 households surveyed
8. Compute Test Statistic Computed Test Stat.= -2
9. Make Statistical Decision Reject Null Hypothesis
10. Express Decision The true mean # of TV set is less than 3 in the US households.
Hypothesis Testing: Steps
Test the Assumption that the average # of TV sets in US homes is at least 3.
(continued)
Lesson. 5 - 17
• Convert Sample Statistic (e.g., ) to Standardized Z Variable
• Compare to Critical Z Value(s) If Z test Statistic falls in Critical Region, Reject H0;
Otherwise Do Not Reject H0
3. Z-Test Statistics (Known)
Test Statistic
X
n
XXZ
X
X
Lesson. 5 - 18
• Assumptions Population Is Normally Distributed If Not Normal, use large samples Null Hypothesis Has or Sign Only
• Z Test Statistic:
One-Tail Z Test for Mean (Known)
n
xxz
x
x
Lesson. 5 - 19
Z0
Reject H0
Z0
Reject H0
H0: H1: < 0
H0: 0 H1: > 0
Must Be Significantly Below = 0
Small values don’t contradict H0
Don’t Reject H0!
Rejection Region
Lesson. 5 - 20
Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.
368 gm.
Example: One Tail Test
H0: 368 H1: > 368
_
Lesson. 5 - 21
Z .04 .06
1.6 .5495 .5505 .5515
1.7 .5591 .5599 .5608
1.8 .5671 .5678 .5686
.5738 .5750
Z0
Z = 1
1.645
.50 -.05
.45
.05
1.9 .5744
Standardized Normal Probability Table (Portion)
What Is Z Given = 0.05?
= .05
Finding Critical Values: One Tail
Critical Value = 1.645
Lesson. 5 - 22
= 0.05
n = 25
Critical Value: 1.645
Test Statistic:
Decision:
Conclusion:
Do Not Reject at = .05
No Evidence True Mean Is More than 368Z0 1.645
.05
Reject
Example Solution: One Tail
H0: 368 H1: > 368 50.1
n
XZ
Lesson. 5 - 23
Does an average box of cereal contains 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.
368 gm.
Example: Two Tail Test
H0: 368
H1: 368
Lesson. 5 - 24
= 0.05
n = 25
Critical Value: ±1.96
Test Statistic:
Decision:
Conclusion:
Do Not Reject at = .05
No Evidence that True Mean Is Not 368Z0 1.96
.025
Reject
Example Solution: Two Tail
-1.96
.025
H0: 386
H1: 38650.1
2515
3685.372
n
XZ
Lesson. 5 - 25
Connection to Confidence Intervals
For X = 372.5oz, = 15 and n = 25,
The 95% Confidence Interval is:
372.5 - (1.96) 15/ 25 to 372.5 + (1.96) 15/ 25
or
366.62 378.38
If this interval contains the Hypothesized mean (368), we do not reject the null hypothesis.
It does. Do not reject.
_
Lesson. 5 - 26
Assumptions Population is normally distributed If not normal, only slightly skewed & a
large sample taken
Parametric test procedure
t test statistic
t-Test: Unknown
nSX
t
Lesson. 5 - 27
Example: One Tail t-Test
Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, andS=15. Test at the 0.01 level.
368 gm.
H0: 368 H1: 368
is not given,
Lesson. 5 - 28
= 0.01
n = 36, df = 35
Critical Value: 2.4377
Test Statistic:
Decision:
Conclusion:
Do Not Reject at = .01
No Evidence that True Mean Is More than 368Z0 2.4377
.01
Reject
Example Solution: One Tail
H0: 368 H1: 368 80.1
3615
3685.372
nSX
t
Lesson. 5 - 29
• Involves categorical variables
• Fraction or % of population in a category
• If two categorical outcomes, binomial
distribution Either possesses or doesn’t possess the characteristic
• Sample proportion (ps)
4. Proportions
sizesample
successesofnumber
n
Xps
Lesson. 5 - 30
Example:Z Test for Proportion
•Problem: A marketing company claims that it receives 4% responses from its Mailing.
•Approach: To test this claim, a random sample of 500 were surveyed with 25 responses.
•Solution: Test at the = .05 significance level.
Lesson. 5 - 31
= .05
n = 500
Do not reject at Do not reject at = .05
Z Test for Proportion: Solution
H0: p .04
H1: p .04
Critical Values: 1.96
Test Statistic:
Decision:
Conclusion:We do not have sufficient
evidence to reject the company’s claim of 4% response rate.
Z p - p
p (1 - p)n
s=
.04 -.05.04 (1 - .04)
500
= -1.14
Z0
Reject Reject
.025.025