Teaching Statistical Concepts with Simulated Data

Andrej Blejec

National Institute of Biology andUniversity of LjubljanaLjubljana, Slovenia

andrej.blejec@nib.si

Kinds of data

• real life data• invented data• simulated data

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Real life data

• interesting for students• need subject matter knowledge to interpret

statistical results• oversimplified problems• complex problems• unclear relevance of statistical results

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Invented data

• a collection of data, no story• useful mostly for calculation drill• no possible interpretation of results• uninteresting for students

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Simulated data

• sampled from known distribution• composed according to prespecified model

e.g. Y = a + b * X + , with known distribution of X and

• known statistical propertiese.g. mean, variance, regression coefficient

• analysis results can be compared to true values

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Goals of teaching statistics

• interest for statistics• understanding of statistical methods• skills in use of statistics• experiences in analyzing data• other ( insert your own :) ___________

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

real life data

invented data

simulated data

interest

understanding

skills

experiences

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Sun

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Absorption spectrum

Sun

Hydrogen

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Sun

Hydrogen

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Least squares estimate of the mean

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Mea

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Mea

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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True mean value μTrue mean value μ

LS estimateLS estimate

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Mea

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Maximum likelihood estimation of the mean

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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True mean value μTrue mean value μ

ML estimateML estimate

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Maximum likelihood estimate of standard deviation

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

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ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

p(x|

sd)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

sd

Log

likel

ihoo

d

0 1 2 3 4 5 6

-100

-60

-20

Log

likel

ihoo

dp

( x

| sd

)

ML estimateML estimate

True standard deviation σTrue standard deviation σ

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Hypothesis testing:

significance

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Hypothesis testing:

significance

or rather a surprise ?

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

020

4060

8010

012

0

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

H0: mu= 50

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

020

4060

8010

012

0

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

H0: mu= 50

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

020

4060

8010

012

0

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

H0: mu= 50

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

020

4060

8010

012

0

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

H0: mu= 50

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

020

4060

8010

012

0

SAMPLE

H0: mu= 50

020

4060

8010

012

0

SAMPLE

H1: mu= 80

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

020

4060

8010

012

0

SAMPLE

H0: mu= 50

020

4060

8010

012

0

SAMPLE

H1: mu= 80

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Confidence intervals

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Weldon, K.L. Statistics: A Conceptual Approach.. Prentice-Hall. New York. 1986

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Bad sampling implies bias

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Bias due to “leave max out”

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Confidence intervals for variance

Biased and unbiased estimators

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 5 10 15 20 25

02

04

06

08

01

00

rnorm | var | n= 5 | biased

var

0 1 2 3 4 5

02

04

06

08

01

00

0 1 2 3 4 5

02

46

81

01

2

var var

0 1 2 3 4 5

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 5 10 15 20 25

02

04

06

08

01

00

rnorm | var | n= 5 |

var

0 1 2 3 4 5

02

04

06

08

01

00

0 1 2 3 4 5

02

46

81

0

var var

0 1 2 3 4 5

02

04

06

08

01

00

unbiased

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 2 4 6

02

04

06

08

01

00

rnorm | var | n= 10 |

var

0 1 2 3 4 5

02

04

06

08

01

00

0 1 2 3 4 5

05

10

15

var var

0 1 2 3 4 5

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 1 2 3 4

02

04

06

08

01

00

rnorm | var | n= 25 |

var

0 1 2 3 4 5

02

04

06

08

01

00

0 1 2 3 4 5

05

10

15

var var

0 1 2 3 4 5

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0.0 0.5 1.0 1.5 2.0

02

04

06

08

01

00

rnorm | var | n= 50 |

var

0 1 2 3 4 5

02

04

06

08

01

00

0 1 2 3 4 5

05

10

15

20

25

var var

0 1 2 3 4 5

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0.0 0.5 1.0 1.5 2.0

02

04

06

08

01

00

rnorm | var | n= 100 |

var

0 1 2 3 4 5

02

04

06

08

01

00

0 1 2 3 4 5

01

02

03

0

var var

0 1 2 3 4 5

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0.0 0.5 1.0 1.5

02

04

06

08

01

00

rnorm | var | n= 200 |

var

0 1 2 3 4 5

02

04

06

08

01

00

0 1 2 3 4 5

01

02

03

04

0

var var

0 1 2 3 4 5

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Confidence intervals for variance:

what if assumptions are not met?

Case of asymetric parent population

var

0 100 200 300 400

02

04

06

08

01

00

rlnorm | var | n= 5 |

var

0 5 10 15 20

02

04

06

08

01

00

0 5 10 15 20

01

02

03

04

0

var var

0 5 10 15 20

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 50 100 150

02

04

06

08

01

00

rlnorm | var | n= 10 |

var

0 5 10 15 20

02

04

06

08

01

00

0 5 10 15 20

05

10

15

20

25

30

var var

0 5 10 15 20

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 50 100 150 200

02

04

06

08

01

00

rlnorm | var | n= 25 |

var

0 5 10 15 20

02

04

06

08

01

00

0 5 10 15 20

05

10

15

20

25

30

var var

0 5 10 15 20

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 10 20 30 40 50

02

04

06

08

01

00

rlnorm | var | n= 50 |

var

0 5 10 15 20

02

04

06

08

01

00

0 5 10 15 20

05

10

15

20

25

var var

0 5 10 15 20

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 5 10 15 20

02

04

06

08

01

00

rlnorm | var | n= 100 |

var

0 5 10 15 20

02

04

06

08

01

00

0 5 10 15 20

05

10

15

20

var var

0 5 10 15 20

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

var

0 5 10 15 20 25 30

02

04

06

08

01

00

rlnorm | var | n= 200 |

var

0 5 10 15 20

02

04

06

08

01

00

0 5 10 15 20

05

10

15

20

25

30

var var

0 5 10 15 20

02

04

06

08

01

00

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Confidence band in linear regression

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5

y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5

y = -0.142 + 0.974 * Xr^2= 0.538

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5

y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5

y = -0.142 + 0.974 * Xr^2= 0.538

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5

y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5

0.153 + 1.07 * Xr^2= 0.413

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 1 )

n= 40 Conf. level = 0.95 r^2 = 0.5

-0.0934 + 0.923 * X

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 1 )

n= 40 Conf. level = 0.95 r^2 = 0.5

-0.0934 + 0.923 * X-0.0934 + 0.923 * X-0.11 + 1.04 * X-0.11 + 1.04 * X-0.00754 + 1.01 * X-0.00754 + 1.01 * X0.0667 + 1.23 * X

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 1 )

n= 40 Conf. level = 0.95 r^2 = 0.5

-0.0934 + 0.923 * X-0.0934 + 0.923 * X-0.11 + 1.04 * X-0.11 + 1.04 * X-0.00754 + 1.01 * X-0.00754 + 1.01 * X0.0667 + 1.23 * X0.0667 + 1.23 * X-0.0458 + 1.18 * X-0.0458 + 1.18 * X0.0297 + 1.08 * X0.0297 + 1.08 * X-0.205 + 0.785 * X-0.205 + 0.785 * X0.0115 + 0.833 * X0.0115 + 0.833 * X0.0933 + 1.2 * X0.0933 + 1.2 * X0.00637 + 0.968 * X0.00637 + 0.968 * X

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 1 )

n= 40 Conf. level = 0.95 r^2 = 0.5

-0.2 + 0.841 * X-0.2 + 0.841 * X-0.257 + 1.18 * X-0.257 + 1.18 * X0.116 + 0.751 * X0.116 + 0.751 * X0.00216 + 0.907 * X

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 1 )

n= 40 Conf. level = 0.95 r^2 = 0.5

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 4 )

n= 40 Conf. level = 0.95 r^2 = 0.2

-0.4 + 0.683 * X

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 4 )

n= 40 Conf. level = 0.95 r^2 = 0.2

-0.4 + 0.683 * X-0.4 + 0.683 * X-0.514 + 1.36 * X-0.514 + 1.36 * X0.232 + 0.501 * X0.232 + 0.501 * X0.00432 + 0.815 * X0.00432 + 0.815 * X-0.695 + 1.04 * X-0.695 + 1.04 * X-0.0815 + 0.528 * X-0.0815 + 0.528 * X-0.234 + 1.07 * X-0.234 + 1.07 * X-0.723 + 0.653 * X-0.723 + 0.653 * X0.161 + 1.56 * X0.161 + 1.56 * X-0.359 + 0.595 * X-0.359 + 0.595 * X

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

x

y

-3 -2 -1 0 1 2 3

-3-2

-10

12

3y = 0 + 1 *N(0, 1 )+N(0, 4 )

n= 40 Conf. level = 0.95 r^2 = 0.2

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

Final remarks

• Graphically supported simulations can to some extent ‑ ‑replace the proofs, usually not understandable to non-mathematics majors. Maybe they are the answer to some questions:

• "If an audience is not convinced by proof, why do proof? (Moore, 1996)

• “Do [students] need to know the theory or do they need to understand the concept?” (J Brown and I David, ICOTS8, 2010)

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011

• Simulated data have to be combined with real life data and projects

• Simulated data can serve as pure and simple data on which we can train our perception for statistical results and learn what patterns and properties in data can be revealed by applied method

• After such preparation students will be able to interpret the real life and subject matter data in all their complexity

Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011