Topic 7 - Chance Variability (Box model)

Question 1

L.O.

Answer 1

LO6 Use the box model to describe chance and chance variability, including sample surveys and the central limit theorem.

Question 2

Chance Variability

Answer 2

Everytime a chance process occurs, chance variability does too.
Observed value = Expected value + chance error

Question 3

Law of Large numbers (Law of averages)

Answer 3

States that proportion of heads becomes more stable as simulation length increases (in coin tossing)

• The chance error in number of heads is large in absolute size, but small relative to the number of tosses

Question 4

The Box model

Answer 4

Simple way to describe many chance processes

Must know:
- The distinct numbers that go into the box (tickets)
- The number of each kind of ‘tickets’ put into box
- The number of draws from the box

Creates a sum or mean of the sampl
- Expected value (EV)
- Observed value (OV)

CHance error = OV - EV

Question 5

Applying the box model

Answer 5

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Question 6

Modelling the sum of a sample

Answer 6

• As the number of draws increases, so does the standard error, in terms of absolute size

Observed value = EV + chance error
where,
EV = # of draws x mean of box
SE = squareroot(# draws) x SD of box

Question 7

Working out the SD of a box

Answer 7

3 ways:
1. Formula; Root of the square mean of the gaps
2. R; popSD()
3. Shortcut for simple binary boxes:
SD = (big-small #) root(proportion of big x prop. of small)

Question 8

Modelling the mean of a sample

Answer 8

For the mean of random draws from a box model with replacement,

OV = EV + CE (chance error)

EV = mean of box

SE = SD box/ Root(# draws)

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Question 9

Binary examples

Answer 9

Easier to make 2 box models
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Question 10

Types of Histograms

Answer 10

Data Histogram:
- Represents # of data by area

Probability Histogram:
- Represents chance by area
- On x-axis, EV measures centre, SE measures spread

Simulation Histogram:
- Represents chance by area, for a simulation of chance processes

Question 11

Central limit theorum

Answer 11

• The sample mean of the large # of random variables is approximately normal
• Larger sample = more normal
• When drawing at random with replacement, if sample size if large enough, simulated curve will follow a normal curve

Question 12

Relationship between Probability and Simulation hitogram

Answer 12

• ## Repeating a simulation histogram will converge to the probability histogram, following the central limit theorum.

Question 13

The continuity correction

Answer 13

To approximate a discrete distribution by the normal distribution, we need to adjust the end points by 0.5. This is called the continuity correction
- To work out whether to add or minus 0.5, draw a sketch of the histogram.

In EXAM:
- it is best to draw it out to determine whether we delete or expand points on a graph.