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Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 1
Advanced Mathematics for
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Advanced Mathematics for Business
Topic 4: Inferential Statistics 2
Inferential Statistics 2 Topic 4 - 4.2
Scope and CoverageThis topic will cover: Hypothesis testing with a sample
- CI, fixed level, significance testing Two sample t-test
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Two sample t test Significance, errors and power Frequency data and the 2 test
Inferential Statistics 2 Topic 4 - 4.3
Learning OutcomesBy the end of this topic students will be able to: Perform a single sample t-test of the mean Perform a two sample t-test Interpret significance probabilities
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Interpret significance probabilities Perform a 2 goodness of fit test
Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 2
Inferential Statistics 2 Topic 4 - 4.4
Hypothesis Testing A Reminder from Topic 3
A car manufacturer releases a new car and claims that its urban cycle fuel efficiency is 18.5 km per litre. A car enthusiast magazine decides to test this claim.
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claim.
- Null hypothesis H0: = 18.5
- Alternative hypothesis is H1: 18.5
Inferential Statistics 2 Topic 4 - 4.5
Hypothesis Testing A Reminder from Topic 3
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Inferential Statistics 2 Topic 4 - 4.6
Hypothesis Testing A Reminder from Topic 3
1 5.00% 2.50% 1.00% 0.50%2 10.00% 5.00% 2.00% 1.00% 90.00% 95.00% 98.00% 99.00% = n - 1
1 6 3138 12 7062 31 8205 63 6567
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1 6.3138 12.7062 31.8205 63.65672 2.9200 4.3027 6.9646 9.92483 2.3534 3.1824 4.5407 5.84094 2.1318 2.7764 3.7469 4.6041
Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 3
Inferential Statistics 2 Topic 4 - 4.7
Hypothesis Testing A Reminder from Topic 3
2.5%2.5%
17.20 18.38
95%
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Inferential Statistics 2 Topic 4 - 4.8
One Sample T-test of the Mean - 1
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Inferential Statistics 2 Topic 4 - 4.9
One Sample T-test of the Mean - 2
2.5%2.5%
-2.7764 2.7764
95%
t(4)
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Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 4
Inferential Statistics 2 Topic 4 - 4.10
One Sample T-test of the Mean - 3
0.5%0.5%
-4.6041 4.6041
99%
t(4)
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Inferential Statistics 2 Topic 4 - 4.11
One-Sided Test
1.0%
-3.7469
99%
t(4)
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Inferential Statistics 2 Topic 4 - 4.12
Significance Level
1.47%1.47%
-3.32 3.32
t(4)
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Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 5
Inferential Statistics 2 Topic 4 - 4.13
The Two Sample T-test Motivation
- To test whether the means of two populations are equal
Assumptions:
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- Both populations essentially normally distributed- Independence of populations- Sample variances differ by less than a factor of 3
Inferential Statistics 2 Topic 4 - 4.14
What Test Statistic?
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Inferential Statistics 2 Topic 4 - 4.15
The Two Sample T-test
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Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 6
Inferential Statistics 2 Topic 4 - 4.16
Example - 1 Two machines on a production line are set to
produce the same part. Workers down the line are concerned that the two machines are not producing parts of the same size. Test the hypothesis.H : = 0 H : 0
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H0: 1 2 = 0 H1: 1 2 0 Sample machine 1: 28.54, 28.27, 25.81, 28.41,
29.09, 28.09, 29.72, 23.56 Sample machine 2: 31.85, 29.96, 31.97, 33.68,
23.81, 30.79
Inferential Statistics 2 Topic 4 - 4.17
Example - 2
2.5%2.5%
-2.1788 2.1788
95%
t(12)
5.0%
-1.7823
95%
t(12)
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1 5.00% 2.50% 1.00% 0.50%2 10.00% 5.00% 2.00% 1.00% 90.00% 95.00% 98.00% 99.00% = n1 + n2 - 2
11 1.7959 2.2010 2.7181 3.105812 1.7823 2.1788 2.6810 3.054513 1.7709 2.1604 2.6503 3.0123
Inferential Statistics 2 Topic 4 - 4.18
Example - 3
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Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 7
Inferential Statistics 2 Topic 4 - 4.19
Errors, Significance and PowerActual
H0 H1
Dec
isio
n H0 Type 2H1 Type 1
= 5%
z = 1 6449
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ActualH0 H1
Dec
isio
n H0 1 - H1 1 -
significance
power
Distribution of Sample Means
z = 1.6449
Inferential Statistics 2 Topic 4 - 4.20
2 Goodness-of-fit Test Motivation
- Decide reasonableness of using particular probability model to describe empirical data
Example uses
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p- Does Poisson probability distribution fit empirical
data?- Are two categorical variables independent?
Inferential Statistics 2 Topic 4 - 4.21
H0: Distribution is Poisson - 1
Number Frequency
0 101 172 42
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2 423 344 125 5
>5 0120
Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 8
Inferential Statistics 2 Topic 4 - 4.22
H0: Distribution is Poisson - 2
Number Frequencyobserved
0 101 172 42
Frequencyexpected
0.1003 12.03110.2306 27.67140 2652 31 8222
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2 423 344 125 5
>5 0120
0.2652 31.82220.2033 24.39700.1169 14.02830.0538 6.45300.0300 3.5971
1 120
Inferential Statistics 2 Topic 4 - 4.23
H0: Distribution is Poisson - 3
Number Frequencyobserved
Frequencyexpected
0 10 0.1003 12.03111 17 0.2306 27.67142 42 0 2652 31 8222
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2 42 0.2652 31.82223 34 0.2033 24.39704 12 0.1169 14.0283
>4 5 0.0838 10.0501120 1 120
Inferential Statistics 2 Topic 4 - 4.24
Critical Values of 2 Distributionsignificance level
p = 0.05 p = 0.01R = 5.00% R = 1.00% = k - m -1
1 3 841 6 635R
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1 3.841 6.6352 5.991 9.2103 7.815 11.3454 9.488 13.2775 11.070 15.086
etc.
R
2
Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 9
Inferential Statistics 2 Topic 4 - 4.25
H0: Distribution is PoissonNumber Frequency
observedFrequencyexpected
(O E)2/E
0 10 0.1003 12.0311 0.341 17 0.2306 27.6714 4.122 42 0.2652 31.8222 3.263 34 0.2033 24.3970 3.784 12 0.1169 14.0283 0.29
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4 12 0.1169 14.0283 0.29>4 5 0.0838 10.0501 2.54
120 1 120 14.32
H0 is rejected at the 5% significance level.
Inferential Statistics 2 Topic 4 - 4.26
2 Test of Association - 1Motivation Is there an association between two categorical
variables?
Method of Follow-up
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Method of Follow-up
email phone visit
satisfied 15 22 17 54
neither 6 6 3 15
dissatisfied 9 2 10 21
30 30 30 90
Inferential Statistics 2 Topic 4 - 4.27
2 Test of Association - 2 Is there an association between two categorical
variables? What are expected counts with no association?
Method of Follow-up
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Method of Follow up
email phone visit
90
Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 10
Inferential Statistics 2 Topic 4 - 4.28
2 Test of Association - 3 Is there an association between two categorical
variables? What are expected counts with no association?
Method of Follow-up
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satisfied
neither
dissatisfied
90
Inferential Statistics 2 Topic 4 - 4.29
2 Test of Association - 4 Is there an association between two categorical
variables? What are expected counts with no association?
Method of Follow-up
email phone visit
NCC Education LimitedV1.0
satisfied
neither
dissatisfied
90
Inferential Statistics 2 Topic 4 - 4.30
2 Test of Association - 5 Observed and expected counts.
Method of Follow-up
email phone visit
satisfied 15 (18) 22 (18) 17 (18) 54
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neither 6 (5) 6 (5) 3 (5) 15
dissatisfied 9 (7) 2 (7) 10 (7) 21
30 30 30 90
Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 11
Inferential Statistics 2 Topic 4 - 4.31
Sampling Frames (probability) Simple random sampling
Systematic random sampling
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Stratified random sampling
Cluster (area) sampling
Inferential Statistics 2 Topic 4 - 4.32
Sampling Frames (non-probability) Quota sampling
Availability / convenience sampling
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Purposive sampling
Inferential Statistics 2 Topic 4 - 4.33
RecapBy the end of this topic students will be able to: Perform a single sample t-test of the mean Perform a two sample t-test Interpret significance probabilities
NCC Education LimitedV1.0
Interpret significance probabilities Perform a 2 goodness of fit test
Topic 4 Inferential Statistics 2 Advanced Mathematics for Business
V1.0 Visuals Handout Page 12
Inferential Statistics 2 Topic 4 - 4.34
Bibliography Buglear, J. Quantitative Methods for Business.
Elsevier Butterworth Heinemann. Hinton, PR. Statistics Explained. Routledge. Thomas, AB. Research Skills for Management
Studies Routledge
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Studies. Routledge. Thomas, AB. Research Concepts for Management
Studies. Routledge. Wisniewski, M. Quantitative Methods for Decision
Makers. FT Prentice Hall.
Inferential Statistics 2 Topic 4 - 4.35
Topic 4 Inferential Statistics 2
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Any Questions?