abm - lecture 04 - inferential statisitcs 2 (1)

Topic 4 Inferential Statistics 2 Advanced Mathematics for Business

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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


Two sample t test Significance, errors and power Frequency data and the 2 test


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


Interpret significance probabilities Perform a 2 goodness of fit test




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.


claim.

- Null hypothesis H0: = 18.5

- Alternative hypothesis is H1: 18.5






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


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





2.5%2.5%

17.20 18.38

95%



One Sample T-test of the Mean - 1




2.5%2.5%

-2.7764 2.7764

95%

t(4)






0.5%0.5%

-4.6041 4.6041

99%

t(4)



One-Sided Test

1.0%

-3.7469

99%

t(4)



Significance Level

1.47%1.47%

-3.32 3.32

t(4)





The Two Sample T-test Motivation

- To test whether the means of two populations are equal

Assumptions:


- Both populations essentially normally distributed- Independence of populations- Sample variances differ by less than a factor of 3


What Test Statistic?



The Two Sample T-test





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


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


Example - 2

2.5%2.5%

-2.1788 2.1788

95%

t(12)

5.0%

-1.7823

95%

t(12)


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


Example - 3





Errors, Significance and PowerActual

H0 H1

Dec

isio

n H0 Type 2H1 Type 1

= 5%

z = 1 6449


ActualH0 H1

Dec

isio

n H0 1 - H1 1 -

significance

power

Distribution of Sample Means

z = 1.6449


2 Goodness-of-fit Test Motivation

- Decide reasonableness of using particular probability model to describe empirical data

Example uses


p- Does Poisson probability distribution fit empirical

data?- Are two categorical variables independent?


H0: Distribution is Poisson - 1

Number Frequency

0 101 172 42


2 423 344 125 5

>5 0120





Number Frequencyobserved

0 101 172 42

Frequencyexpected

0.1003 12.03110.2306 27.67140 2652 31 8222


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



Number Frequencyobserved

Frequencyexpected

0 10 0.1003 12.03111 17 0.2306 27.67142 42 0 2652 31 8222


2 42 0.2652 31.82223 34 0.2033 24.39704 12 0.1169 14.0283

>4 5 0.0838 10.0501120 1 120


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


1 3.841 6.6352 5.991 9.2103 7.815 11.3454 9.488 13.2775 11.070 15.086

etc.

R

2




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


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.


2 Test of Association - 1Motivation Is there an association between two categorical

variables?

Method of Follow-up


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


2 Test of Association - 2 Is there an association between two categorical

variables? What are expected counts with no association?

Method of Follow-up


Method of Follow up

email phone visit

90






Method of Follow-up


satisfied

neither

dissatisfied

90




Method of Follow-up

email phone visit


satisfied

neither

dissatisfied

90


2 Test of Association - 5 Observed and expected counts.

Method of Follow-up

email phone visit

satisfied 15 (18) 22 (18) 17 (18) 54


neither 6 (5) 6 (5) 3 (5) 15

dissatisfied 9 (7) 2 (7) 10 (7) 21

30 30 30 90




Sampling Frames (probability) Simple random sampling

Systematic random sampling


Stratified random sampling

Cluster (area) sampling


Sampling Frames (non-probability) Quota sampling

Availability / convenience sampling


Purposive sampling


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


Interpret significance probabilities Perform a 2 goodness of fit test




Bibliography Buglear, J. Quantitative Methods for Business.

Elsevier Butterworth Heinemann. Hinton, PR. Statistics Explained. Routledge. Thomas, AB. Research Skills for Management

Studies Routledge


Studies. Routledge. Thomas, AB. Research Concepts for Management

Studies. Routledge. Wisniewski, M. Quantitative Methods for Decision

Makers. FT Prentice Hall.


Topic 4 Inferential Statistics 2


Any Questions?

abm - lecture 04 - inferential statisitcs 2 (1)

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