Introduction Hypothesis Testing
-
Upload
muhammad-ali-gunawan -
Category
Education
-
view
2.196 -
download
2
description
Transcript of Introduction Hypothesis Testing
![Page 1: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/1.jpg)
Introduction to Hypothesis Testing:
One Population Value
Chapter 8 Handout
![Page 2: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/2.jpg)
Chapter 8 Summary
Hypothesis Testing for One Population Value:1. Population Mean (
a. (population standard deviation) is given (known): Use z/standard normal/bell shaped distribution
b. (pop std dev) is not given but s (sample std dev) is given Use student’s t distribution
2. Population proportion () Use z/standard normal/bell shaped distribution
3. Population variance ( Use (Chi-Square) distribution
PS: population standard deviation =
![Page 3: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/3.jpg)
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.
What is a Hypothesis?
![Page 4: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/4.jpg)
• 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
•Refers to the Status Quo•Always contains the ‘ = ‘ sign
•The Null Hypothesis may or may not be rejected.
![Page 5: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/5.jpg)
• Is the opposite of the null hypothesise.g. The average # TV sets in US homes
is less than 3 (H1: < 3)
• Challenges the Status Quo
• Never contains the ‘=‘ sign
• The Alternative Hypothesis may or may not be accepted
The Alternative Hypothesis, H1
or HA
![Page 6: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/6.jpg)
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
![Page 7: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/7.jpg)
Population
Assume thepopulationmean age is 50.(Null Hypothesis)
REJECT
The SampleMean Is 20
SampleNull Hypothesis
50?20 XIs
Hypothesis Testing Process
No, not likely!
![Page 8: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/8.jpg)
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
![Page 9: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/9.jpg)
• Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling
Distribution
• Designated a(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, a
![Page 10: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/10.jpg)
Level of Significance, aand the Rejection Region
H0: 3
H1: < 30
0
0
H0: 3
H1: > 3
H0: 3
H1: 3
a
a
a/2
Critical Value(s)
Rejection Regions
![Page 11: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/11.jpg)
• Type I Error Reject True Null Hypothesis Has Serious Consequences Probability of Type I Error Is a
Called Level of Significance
• Type II Error Do Not Reject False Null Hypothesis Probability of Type II Error Is b (Beta)
Errors in Making Decisions
![Page 12: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/12.jpg)
H0: Innocent
Jury Trial Hypothesis Test
Actual Situation Actual Situation
Verdict Innocent Guilty Decision H0 True H0 False
Innocent Correct ErrorDo NotReject
H0
1 - a Type IIError (b )
Guilty Error Correct RejectH0
Type IError(a )
Power(1 - b)
Result Possibilities
![Page 13: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/13.jpg)
a
b
Reduce probability of one error and the other one goes up.
a& bHave an Inverse Relationship
![Page 14: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/14.jpg)
• True Value of Population Parameter Increases When Difference Between Hypothesized
Parameter & True Value Decreases
• Significance Level a Increases When aDecreases
• Population Standard Deviation Increases When Increases
• Sample Size n Increases When n Decreases
Factors Affecting Type II Error, b
a
b
b
b
n
![Page 15: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/15.jpg)
3 Methods for Hypotheses Tests
Refer to Figure 8-6 (page 299) for a hypothesis test for means with (pop. std. dev.) is given:
Method 1: Comparing XaX critical) with XMethod 2: Z test, i.e., comparing Za critical) with Z (or Z statistics or Z
calculated)Method 3: Comparing asignificance level) with p-valueYou can modify those three methods for other cases. For example, if is
unknown, you must use student’s t distribution. If you would like to use Method 2, please compare t at critical) with t (or t statistics or t calculated). Refer to Figure 8-8 (page 303).
![Page 16: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/16.jpg)
You always get: • Za critical) from Z distribution• tat critical) from student’s t distribution• .acritical) from distributionYou always get:• Z or Z calculated or Z statistics from sample (page 299 and Figure 8-6)• t or t calculated or t statistics from sample (Figure 8-8, page 299)• .or calculated or statistics from sample (Figure 8-19, page 322)
![Page 17: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/17.jpg)
• 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
Z-Test Statistics (Known)
Test Statistic
X
n
XXZ
X
X
![Page 18: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/18.jpg)
• Probability of Obtaining a Test Statistic More Extreme or ) than Actual Sample Value Given H0 Is True
• Called Observed Level of Significance Smallest Value of a H0 Can Be Rejected
• Used to Make Rejection Decision If p value a Do Not Reject H0
If p value <a, Reject H0
p Value Test
![Page 19: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/19.jpg)
1. State H0 H0 : 3
2. State H1 H1 :
3. Choosea a = .05
4. Choose n n = 100
5. Choose Method: Z Test (Method 2)
Hypothesis Testing: Steps
Test the Assumption that the true mean # of TV sets in US homes is at least 3.
![Page 20: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/20.jpg)
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)
![Page 21: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/21.jpg)
• 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
![Page 22: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/22.jpg)
Z0
a
Reject H0
Z0
Reject H0
a
H0: H1: < 0
H0: 0 H1: > 0
Must Be Significantly Below = 0
Small values don’t contradict H0
Don’t Reject H0!
Rejection Region
![Page 23: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/23.jpg)
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 a0.05 level.
368 gm.
Example: One Tail Test
H0: 368 H1: > 368
_
![Page 24: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/24.jpg)
Z .04 .06
1.6 .4495 .4505 .4515
1.7 .4591 .4599 .4608
1.8 .4671 .4678 .4686
.4738 .4750
Z0
Z = 1
1.645
.50 -.05
.45
.05
1.9 .4744
Standardized Normal Probability Table (Portion)
What Is Z Givena = 0.05?
a = .05
Finding Critical Values: One Tail
Critical Value = 1.645
![Page 25: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/25.jpg)
a= 0.025
n = 25
Critical Value: 1.645
Test Statistic:
Decision:
Conclusion:
Do Not Reject Ho ata = .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
![Page 26: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/26.jpg)
Z0 1.50
p Value.0668
Z Value of Sample Statistic
From Z Table: Lookup 1.50
.9332
Use the alternative hypothesis to find the direction of the test.
1.0000 - .9332 .0668
p Value is P(Z 1.50) = 0.0668
p Value Solution
![Page 27: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/27.jpg)
0 1.50 Z
Reject
(p Value = 0.0668) (a = 0.05). Do Not Reject.
p Value = 0.0668
a= 0.05
Test Statistic Is In the Do Not Reject Region
p Value Solution
![Page 28: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/28.jpg)
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 a0.05 level.
368 gm.
Example: Two Tail Test
H0: 368
H1: 368
![Page 29: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/29.jpg)
a= 0.05
n = 25
Critical Value: ±1.96
Test Statistic:
Decision:
Conclusion:
Do Not Reject Ho at a = .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
![Page 30: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/30.jpg)
Two tail hypotheses tests = 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 Ho.
_
![Page 31: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/31.jpg)
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
![Page 32: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/32.jpg)
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 a0.01 level.
368 gm.
H0: 368 H1: 368
is not given,
![Page 33: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/33.jpg)
a= 0.01
n = 36, df = 35
Critical Value: 2.4377
Test Statistic:
Decision:
Conclusion:
Do Not Reject Ho at a = .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
![Page 34: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/34.jpg)
• 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 (p)
Proportions
sizesamplesuccessesofnumber
nXp
![Page 35: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/35.jpg)
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 n = 500 were surveyed with x = 25 responses.
•Solution: Test at the a = .05 significance level.
![Page 36: Introduction Hypothesis Testing](https://reader033.fdocument.org/reader033/viewer/2022061111/5454e86ab1af9f33608b45e2/html5/thumbnails/36.jpg)
a = .05n = 500, x = 25 p = x/n = 25/500 = 0.05
Do not reject Ho at Do not reject Ho at a = .05
Z Test for Proportion: Solution
H0: .04
H1: .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-
(1 - )n
=.05-.04
.04 (1 - .04)500
= 1.14
Z0
Reject Reject
.025.025