Schaum

1
Schaums Outline Probability and
Statistics Chapter 7 HYPOTHESIS
TESTING presented by Professor Carol
Dahl Examples by Alfred Aird Kira Jeffery
Catherine Keske Hermann Logsend Yris Olaya
2
Outline of Topics
Topics Covered

• Statistical Decisions
• Statistical Hypotheses
• Null Hypotheses
• Tests of Hypotheses
• Type I and Type II Errors
• Level of Significance
• Tests Involving the Normal Distribution
• One and Two Tailed Tests
• P Value

3
Outline of Topics (Continued)

• Special Tests of Significance
• Large Samples
• Small Samples
• Estimation Theory/Hypotheses Testing
  Relationship
• Operating Characteristic Curves and Power of a
  Test
• Fitting Theoretical Distributions to Sample
  Frequency Distributions
• Chi-Square Test for Goodness of Fit

4
The Truth Is Out ThereThe Importance of
Hypothesis Testing

• Hypothesis testing
• helps evaluate models based upon real data
• enables one to build a statistical model
• enhances your credibility as
• analyst
• economist

5
Statistical Decisions

• Innocent until proven guilty principle
• Want to prove someone is guilty
• Assume the opposite or status quo - innocent
• Ho Innocent
• H1 Guilty
• Take subsample of possible information
• If evidence not consistent with innocent - reject
• Person not pronounced innocent but not guilty

6
Statistical Decisions

• Status quo innocence null hypothesis
• Evidence sample result
• Reasonable doubt confidence level

7
Statistical Decisions

• Eg. Tantalum ore deposit
• feasible if quality gt 0.0600g/kg with 99
  confidence
• 100 samples collected from large deposit at
  random.
• Sample distribution
• mean of 0.071g/kg
• standard deviation 0.0025g/kg.

8
Statistical Decisions

• Should the deposit be developed?
• Evidence 0.071 (sample mean)
• Reasonable doubt 99
• Status quo do not develop the deposit
• Ho ? lt 0.0600
• H1 ? gt 0.0600

9
Statistical Hypothesis

• General Principles
• Inferences about population using sample
  statistic
• Prove A is true by assuming it isnt true
• Results of experiment (sample) compared with
  model
• If results of model unlikely, reject model
• If results explained by model, do not reject

10
Statistical Hypothesis
Event A fairly likely, model would be
retained Event B unlikely, model would be
rejected
11
Statistical Decisions

• Should the deposit be developed?
• Evidence 0.071 (sample mean)
• Reasonable doubt 99
• Status quo do not develop the deposit
• Ho ? 0.0600
• H1 ? gt 0.0600
• How likely Ho given 0.071

12
Need Sampling Statistic

• Need statistic with
• population parameter
• estimate for population parameter
• its distribution

13
Need Sampling Statistic

• Population Normal - Two Choices
• Small Sample lt30
• Known Variance
  Unknown Variance

N(0,1) tn-1
14
Need Sampling Statistic

• Population Not-Normal
• Large Sample
• Known Variance Unknown Variance

N(0,1) N(0,1)

Doesnt matter if know variance of not If
population is finite sampling no replacement
need adjustment
15
Normal Distribution
27
XN(0,1)
? 0
SD1 (68)
SD2 (95)
SD3 (99.7)
16
Statistical Decisions

• Should the deposit be developed?
• Evidence 0.071 (sample mean)
• 0.0025g/kg (sample
  variance)
• 0.05 (sample standard
  deviation)
• Reasonable doubt 99
• Status quo do not develop the deposit
• Ho ? 0.0600
• H1 ? gt 0.0600
• One tailed test
• How likely Ho given 0.071

17
Hypothesis test

• Evidence 0.071 (sample mean)
• 0.05g/kg (sample standard
  deviation)
• Reasonable doubt 99
• Status quo do not develop the deposit
• Ho ? 0.0600
• H1 ? gt 0.0600

18
Statistical Hypothesis

Eg. Z (0.071 0.0600)/ (0.05/ ? 100)
2.2 Conclusion Dont reject Ho ,
dont develop deposit
Zc2.33
2.2
19
Null Hypothesis

• Hypotheses cannot be proven
• reject or fail to reject
• based on likelihood of event occurring
• null hypothesis is not accepted

20
Test of Hypotheses Maple Creek Mine and Potaro
Diamond field in Guyana

• Mine potential for producing large diamonds
• Experts want to know true mean carat size
  produced
• True mean said to be 4 carats
• Experts want to know if true with 95 confidence
• Random sample taken
• Sample mean found to be 3.6 carats
• Based on sample, is 4 carats true mean for mine?

21
Tests of Hypotheses

• Tests referred to as
• Tests of Hypotheses
• Tests of Significance
• Rules of Decision

22
Types of Errors
Ho µ 4 (Suppose this is true) H1 µ ? 4 Two
tailed test Choose ? 0.05 Sample n 100
(assume X is normal), ? 1
23
Type I error (?) reject true
Ho µ 4 suppose true
?/2
?/2
24
Type II Error (ß) - Accept False

• Ho µ 4 not true
• µ 6 true
• ?X-µ not mean 0 but mean 2

ß
25
Lower Type I What happens to Type II

• Ho µ 4 not true
• µ 6 true

ß
26
Higher µWhat happens to Type II?

• Ho µ 4 not true
• µ 7 true
• ?X-µ not mean 0 but mean 3

ß
27
Type I and Type II Errors

• Two types of errors can occur in hypothesis
  testing
• To reduce errors, increase sample size when
  possible

Ho True Ho False
Reject Ho Type I Error Correct Decision
Do Not Reject Ho Correct Decision Type II Error
28
To Reduce Errors

• Increase sample size when possible
• Population, n 5, 10, 20

29
Error Examples

• Type I Error rejecting a true null hypothesis
• Convicting an innocent person
• Rejecting true mean carat size is 4 when it is
• Type II Error not rejecting a false null
  hypothesis
• Setting a guilty person free
• Not rejecting mean carat size is 4 when its not

30
Level of Significance (?)

• a max probability were willing to risk Type I
  Error
• tail area of probability density function
• If Type I Errors cost high, choose a low
• a defined before hypothesis test conducted
• a typically defined as 0.10, 0.05 or 0.01
• a 0.10 for 90 confidence of correct test
  decision
• a 0.05 for 95 confidence of correct test
  decision
• a 0.01 for 99 confidence of correct test
  decision

31
Diamond Hypothesis Test Example
Ho µ 4 H1 µ ? 4 Choose a 0.01 for 99
confidence Sample n 100, ? 1 ?X 3.6,
-Zc - 2.575, Zc 2.575

-2.575
2.575
.005
.005
32
21
Example Continued
1

• Observed not significantly different from
  expected
• Fail to reject null hypothesis
• Were 99 confident true mean is 4 carats

33
Tests Involving the t Distribution

• Billy Ray has inherited large, 25,000 acre
  homestead
• Located on outskirts of Murfreesboro, Arkansas,
  near
• Crater of Diamonds State Park
• Prairie Creek Volcanic Pipe
• Land now used for
• agricultural
• recreational
• No official mining has taken place

34
Case Study in Statistical Analysis Billy Rays
Inheritance

• Billy Ray must now decide upon land usage
• Options
• Exploration for diamonds
• Conservation
• Land biodiversity and recreation
• Agriculture and recreation
• Land development

35
Consider Costs and Benefits of Mining

• Cost and Benefits of Mining
• Opportunity cost
• Excessive diamond exploration damages lands
  value
• Exploration and Mining Costs
• Benefit
• Value of mineral produced

36
Consider Costs and Benefits of Mining

• Cost and Benefits of Mining
• Sample for geologic indicators for diamonds
• kimberlite or lamporite
• larger sample more likely to represent true
  population
• larger sample will cost more

37
How to decide one tailed or two tailed

• One tailed test
• Do we change status quo only if its bigger than
  null
• Do we change status quo only if its smaller than
  null
• Two tailed test
• Change status quo if its bigger of if it smaller

38
Tests of Mean

• Normal or t
• population normal
• known variance
• small sample

Normal
population normal unknown variance small sample
t
large population
Normal
39
Difference Normal and t
t fatter tail than normal bell-curve
40
Hypothesis and Sample

• Need at least 30 g/m3 mine
• Null hypothesis Ho µ 20
• Alternative hypothesis H1 ?
• Sample data n16 (holes drilled)
• X close to normal
• ?X 31 g/m³
• variance (s2/n)0.286 g/m³

41
Normal or t?

• One tailed
• Null hypothesis Ho µ 30
• Alternative hypothesis H1 µ gt 30
• Sample data n 16 (holes drilled)
• ?X 31 g/m³
• variance (s2) 4.29 g/m³ 4.29
• standard deviation s 2.07
• small sample, estimated variance, X close to
  normal
• not exactly t but close if X close to normal

42
Tests Involving the t Distribution

• tn-1 ?X - µ
• s/?n

t16-1
? 0
Reject 5
tc1.75
43
Tests Involving the t Distribution

• tn-1 ?X - µ (31 - 30) 1.93
• s/?n 2.07/ ?16

t16-1
? 0
Reject 5
tc1.75
44
Wells produces oil

• X API Gravity
• approximate normal with mean 37?
• periodically test to see if the mean has changed
• too heavy or too light revise contract
• Ho
• H1
• Sample of 9 wells, ?X 38?, s2 2
• What is test statistic?
• Normal or t?

45
Two tailed t test on mean

• tn-1 ?X - µ
• s/?n

? 0
Reject ?/2
Reject ?/2
tc
tc
46
Two tailed t test on mean

• Ho µ 37
• H1 µ? 37
• Sample of 9 wells, ?X 38?, s2 2, ? 10
• tn-1 ?X - µ (38 37) 1.5
• s/?n 2/ ? 9

47
P-values - one tailed test

• Level of significance for a sample statistic
  under null
• Largest ? for which statistic would reject null
• t16-1 ?X - µ (31 - 30) 1.93
• s/?n 2.07/ ?16

P0.04
tinv(1,87,15,1)
48
P-value two tailed test

• Ho µ 37
• H1 µ? 37
• Sample of 9 wells, ?X 38?, s2 2, ? 10
• tn-1 ?X - µ (38 37) 1.5
• s/?n 2/ ? 9

TDIST(1.5,8,2) 0.172
49
Formal Representation of p-Values

• p-Value lt ? Reject Ho
• p-Value gt ? Fail to reject Ho

50
More tests

• Survey - Ranking refinery managers
• Daily refinery production
• Sample two refineries of 40 and 35 1000
  b/cd
• First refinery mean 74, stand.
  dev. 8
• Second refinery mean 78, stand. dev.
  7
• Questions difference of means?
• variances?
• differences of variances
• Again Statistics Can Help!!!!

51
Differences of Means

• Ho µ1 - µ2 0
• Ho µ1 - µ2 ? 0
• X1 and X2 normal, known variance
• or large sample known variance
• ? 10

5
5
-Zc
Zc
52
Differences of Means

• Ho µ1 - µ2 0
• Ho µ1 - µ2 ? 0
• n1 40, n2 35
• ?X1 74, ?1 8
• ?X2 78, ?2 7

5
5
-Z-1.645c
Zc-1.645
53
Difference of Means

• X normal
• Unknown but equal variances
• Do above test with

54
Variance test (?2 distribution)

Two tailed
?/2
?/2
55
Variance test (?2 distribution)

One tailed
?
56
Hypothesis Test on Variance
Suppose best practice in refinery ?2 6 Does
refinery 2 have different variability than best
practice? Ho ?2 6 H1 ?2 ? 6.5 Example 2nd
mine, n 1 34, Standard deviation 7
57
Hypothesis Test on Variance

• ?/2

Ho ?2 (6.5) 2 H1 ?2 ? 6.52 Example 2nd mine,
n 1 34, Standard deviation 7 ? 10

58
Hypothesis Test on Variance

• ?/2

Suppose best practice in refinery Ho ?2
6.5 H1 ?2 ? 6.5 Example 2nd mine, n 1 34,
Standard deviation 7
59
Variance test (?2 distribution)

Two tailed
0.05
0.05
21.664
48.602
60
Variance test (?2 distribution)

• More variance than best practice

Ho ?2 6.5 H1 ?2 gt 6.5
One tailed
0.10
61
Variance test (?2 distribution)

• More variance than best practice

Ho ?2 6.5 H1 ?2 gt 6.5
One tailed
0.10
chiinv(0.10,34)44.903
62
Testing if Variances the Same F Distribution

• 2 samples of size n1 and n2
• sample variances s12, s22,
• Ho ?12 ?22 gt Ho ?22/?12 1
• Ho ?12 ? ?22 gt Ho ?22/?12 ? 1

63
Testing if Variances the Same F Distribution

• Ho ?12/?22 1
• H1 ?12/?22 ? 1

Two tailed
?/2
?/2
64
Testing if Variances the Same F Distribution

• Ho ?22/?12 1
• H1 ?22/?12gt1

One tailed
?10
65
Example Testing if Variances the Same

• 2 samples of size n1 40
• and n2 35
• sample variances s12 82, s22 72
• Ho ?22/?12 1
• Ho ?22/?12 ? 1

0.579,
1.749
82/721.306
66
Testing if Variances the Same F Distribution

• Ho ?12/?22 1
• H1 ?12/?22? 1

Two tailed
0.05
0.05
Finv(0.05,39,34)1.749
Finv(0.95,39,34)0.579
67
Testing if Variances the Same F Distribution

• Ho ?22/?12 1
• H1 ?22/?12 ? 1

One tailed
0.05
Finv(0.10,39,34)1.544
68
Power of a test

• Type II error
• ? P(Fail to reject Ho H1 is true)
• Power 1- ?

69
Power of a test

• Type II error
• ? P(Fail to reject Ho H1 is true)
• Power 1- ?

70
Power of a test

• Researcher controls level of significance, ?
• Increase ? what happens to ß?

71
Raise Type I (? )What happens to Type II (ß)

• Ho µ 4 not true
• µ 6 true
• ?X-µ not mean 0 but mean 2

ß
72
Higher ?What happens to Type II?
ß
Increase ß, reduce ?
73
Operating Characteristic Curve

Can graph ? against ? called operating
characteristic curve useful in experimental
design
74
Operating Characteristic Curve

75
Fitting a probability distribution

• Is electricity demand a log-normal distribution
• Observed Mean 18.42
• Observed Variance 43
• Observations 20

76
Fitting a probability distribution

• Does electricity demand follow a normal
  distribution?

Observed Mean 18.42 Observed Variance
43 Observations 20
77
You can test your model graphically
1. Order observations from smallest Y1 to
largest Yn 2. Compute cumulative frequency
distribution 3. Plot ordered observations
versus Pi on special probability sheet 4.
If straight line within critical range cant
reject normal
78
You can test your model graphically
9.26 0.05 17.27 0.55
9.83 0.10 18.18 0.60
12.85 0.15 20.28 0.65
13.11 0.20 20.30 0.70
13.23 0.25 20.88 0.75
13.90 0.30 21.98 0.80
14.18 0.35 23.31 0.85
15.99 0.40 24.35 0.90
16.47 0.45 30.24 0.95
17.24 0.50 35.68 1.00
79
Or use the Graph/Probability Plot Option in
Minitab
80
Statistical test of distribution

• Ho Xe? N(µ,?2)
• H1 Xe does not follow N(µ,?2)
• Order data
• Estimate sample mean variance
• Observed Mean 18.42
• Observed Variance 43
• Observations 20
• ?2 statistic goodness of fit of model

81
Statistical test of distribution
Again order sample Create m 5 categories
9.26 17.27
9.83 18.18
12.85 20.28
13.11 20.30
13.23 20.88
13.90 21.98
14.18 23.31
15.99 24.35
16.47 30.24
17.24 35.68
lt10
10-15
15-20
20-25
gt25
82
Statistical test of distribution
9.26 17.27
9.83 18.18
12.85 20.28
13.11 20.30
13.23 20.88
13.90 21.98
14.18 23.31
15.99 24.35
16.47 30.24
17.24 35.68
Actual frequencies
lt10 2
10-15 5
15-20 5
20-25 6
gt25 2
83
Statistical test of distribution
Frequencies
  actual expected
lt10 2  Normdist(10,18.42,6.56,1)20
10-15 5 (Normdist(15,18.42,6.56,1) - Normdist(10,18.42,6.56,1)20
15-20 5 (Normdist(20,18.42,6.56,1) Normdist(15,18.42,6.56,1)20
20-25 6  
gt25 2  
84
Statistical test of distribution
Frequencies
  Observed Expected
lt10 2 1.99
10-15 5 4.03
15-20 5 5.88
20-25 6 4.94
gt25 2 3.16
85
?2 Goodness of Fit Test

• Is based on
• ?2 ?(oi-ei)2/ei

m
i1
df m k 1 k number of parameters replaced
by estimates oi observed frequency, ei expected
frequency
86
Statistical test of distribution
Frequencies
  oi ei
lt10 2 1.99
10-15 5 4.03
15-20 5 5.88
20-25 6 4.94
gt25 2 3.16
?2 ?(oi-ei)2/ei
(2-1.99)2/1.99 (5-4.03)2/4.03 (5-5.88)2/5.88 (
6-4.94)2/4.94 (2-3.19)2/3.16 1.04
87
Statistical test of distribution
Ho X ? N(µ,?2) H1 X does not follow
N(µ,?2) df m k 1 5 2 - 1
?2 ?(oi-ei)2/ei 1.04
CHIINV(0.05,2)5.99
88
Outline of Topics (Continued)

• Estimation Theory/Hypotheses Testing Relationship
• Operating Characteristic Curves and Power of a
  Test
• Fitting Theoretical Distributions to Sample
  Frequency Distributions
• Chi-Square Test for Goodness of Fit

89
Sum Up Chapter 7

• Hypothesis testing
• null vs alternative
• null with equal sign
• null often status quo
• alternative often what want to prove type
  I error vs type II error
• type I called level of significance
• P values
• 1-ß power of test
• probability of rejecting false
• one tailed vs two tailed

90
Sum Up Chapter 7

• Hypothesis tests
• mean Normal test
• population normal, known variance
• large sample
• mean t test
• population normal, unknown variance,
• small sample
• Statistical Decisions
• Statistical Hypotheses
• Null Hypotheses
• Tests of Hypotheses
• Type I and Type II Errors
• Level of Significance
• Tests Involving the Normal Distribution
• One and Two Tailed Tests

91
Sum Up Chapter 7
Normal and t
92
Sum Up Chapter 7

• Hypothesis tests
• difference of means Normal test
• population normal, known variance

93
Sum Up Chapter 7

• Hypothesis tests
• variance

Are variances equal
94
Sum Up Chapter 7
?2 and F
95
Sum Up Chapter 7

• How is random variable distributed
• normal graph cumulative frequency distribution
• special paper
• straight line
• Statistical

?2k-m-1 ?(oi-ei)2/ei k categories m
estimated parameters always 1 tailed
96
End of Chapter 7!