The Box Model - lec PDF

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Summary

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Description

Course Overview Population

3 Sampling Data

1 Exploring Data

Sample

4 Decisions with Data

2 Modelling Data

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ƃ Module3 Sampling Data

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Understanding Chance What is chance?

Chance Variability How can we model chance variability by a box model?

Normal Curve What is the Normal Curve? And what does it have to do with sample mean?

Sample Surveys How can we model the chance variability in sample surveys?

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ĕ The Box Model

w

Data Story: Coin Tossing in WWII The Box Model Modelling the Sum of a sample Modelling the Mean of a sample Summary

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Data Story Coin tossing in WWII

Coin tossing in WWII · John Edmund Kerrich (1903–1985) was a mathematician noted for a series of experiments in probability which he conducted while imprisoned in Nazi-occupied Denmark (Viborg, Midtjylland) in the 1940s. · Kerrich had travelled from South Africa to visit his in-laws in Copenhagen, and arrived just 2 days after Denmark was invaded by Nazi Germany!

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· With a fellow internee Eric Christensen, Kerrich set up a sequence of experiments demonstrating the empirical validity of a number of fundamental laws of probability. - They tossed a (fair) coin 10,000 times and counted the number of heads (5067). - They made 5000 draws from a container with 4 ping pong balls (2x2 different brands), ‘at the rate of 400 an hour, with - need it be stated - periods of rest between successive hours.’ - They investigated tosses of a “biased coin”, made from a wooden disk partly coated in lead. · In 1946 Kerrich published his finding in a monograph An Experimental Introduction to the Theory of Probability.

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 Statistical Thinking

w Kerrich and Christensen tossed a coin 10,000 times. · How many heads would you expect them to get? · How many heads did they actually observe? · Was the difference between expected and observed unusual?

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The Box Model

The box model ? The box model w · The box model is a simple way to describe many chance processes. - We will just be drawing tickets from a box. · We need to know: - the number or proportion of each kind of tickets in the box. - the number of draws from the box. - for now we only consider drawing with replacement.

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Coin tossing in WWII Kerrich and Christensen’s experiment can be described a simple Box Model. 1 (Head); 0 (Tail) 10000 draws  replace

Sample

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Modelling the Sum of a sample

Modelling the Sum of a sample ? Sum of draws from a box model w For the Sum of random draws from a box model with replacement,

where:

ƭ The standard error is the expected magnitude of the chance error. w

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How to work out the SD of the box · As the box represents the population, the SD of the box is the population SD. · We could call it

, but in this context will will simply use

.

3 ways to calculate the SD of the box 1. Formula: RMS(gaps) = Root of the Mean of the Squared gaps. 2. R: popsd() with package multicon 3. Short cut (for simple binary (two tickets) boxes)

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? Short cut for SD of binary box

w If a box only contains 2 different numbers (“big” and “small”), then

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How does chance error relate to standard error? · An observed value is likely to be around its expected value, with a chance error similar to the SE. · Observed values are rarely more than 2 or 3 SEs away from the expected value.

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Example 1: Coin tossing in WWII

Previous thoughts  Statistical Thinking w Kerrich and Christensen tossed a coin 10,000 times. · How many heads would you expect them to get? · How many heads did they actually observe? · Was the difference between expected and observed unusual?

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· How many heads do we expect? [5000 = EV]

Coin tossing in WWII Kerrich and Christensen’s experiment can be described a simple Box Model. 1 (Head); 0 (Tail) 10000 draws  replace

Sample

· How many heads do we expect? [5000 = EV] · How many heads did they observe? [5067 = OV] · What chance error did we expect? [50 = SE] · How big was the chance error? [67 = CE]

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Example 1 again slowly

Step1: Draw the box model 1 (Head); 0 (Tail) 10000 draws  replace

Sample

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Step2: Calculate the mean and SD of the box 1 (Head); 0 (Tail) 10000 draws  replace

Sample

· The mean of the box is ·

The SD of the box is

· Or using the short cut, the SD is

. . .

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Step3: Calculate the EV and SE of the Sum of the sample 1 (Head); 0 (Tail) 10000 draws  replace

Sample

· The mean of the box is 0.5. · The SD of the box is 0.5. · The EV of the Sum of the draws is · The SE of the Sum of the draws is

. .

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Step4: Conclusion · We would expect a sample Sum of 5000 (EV) with SE 50. · Note: We observed a sample Sum of 5067 (OV), that is a chance error 67.

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In R library(multicon) box = c(1, 0) mean(box)

## [1] 0.5

popsd(box)

## [1] 0.5

10000 * mean(box) V E#

## [1] 5000

sqrt(10000) * popsd(box) E S #

## [1] 50

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Modelling the Mean of a sample

Mean of draws from a box model As the Mean of the sample is just the the Sum of the sample divided by the number of the draws, we get an equivalent result as follows.

? Mean of draws from a box model w For the Mean of random draws from a box model with replacement,

where:

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Example2: WWII Coin Tossing (Now with Mean instead of Sum)

Step1: Draw the box model 1 (Head); 0 (Tail) 10000 draws  replace

Sample

Step2: Calculate the mean and SD of the box · The mean of the box is · The SD of the box is

. .

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Step3: Calculate the EV and SE of the Mean of the sample · The EV of the Mean of the draws is · The SE of the Mean of the draws is

. .

Step4: Conclusion · We would expect a sample Mean of 0.5 (EV) with SE 0.005. · Note: We observed a sample Mean of 0.5067 (OV), that is a chance error of 0.0067.

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Summary · The Box Model models a simple chance process involving drawing tickets from a fixed box (population). · We can describe the behaviour of the Sum and the Mean of the Sample in terms of the expected value (EV) and standard error (SE), and compare to the observed value (OV). · We can find

by using popsd() .

· Given the mean and SD of the population: EV Sum of the sample Mean of the sample

mean mean

SE SD SD /

· When there is one desired outcome: Make the desired tickets a “1” and all the other tickets “0”.

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Key Words box model, modelling the mean, modelling the sum, observed value, expected value, chance error, standard error

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STAT 1040--Ch 16 Box Models

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Box Model Help #1 for USU STAT 1040

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