Name the steps of the modelling cycle.

Observe, devise a model, use it for predictions, compare with data, refine, report.

What is an assumption in a model?

A simplification that makes the situation easier to analyse.

Three common probability assumptions?

Outcomes equally likely, trials independent, probability constant.

Expected frequency formula?

np for n trials and probability p.

Relative frequency formula?

Observed frequency divided by the total number of trials.

How does the relative frequency behave as the number of trials grows?

It settles close to the model's probability.

What would suggest a model is unsuitable?

Large or systematic differences between observed and expected values.

What would suggest a model is reasonable?

Small, patternless differences that could be due to chance.

Model for a fair 6-sided die?

P(X=x)=\frac16 for x=1,\ldots,6

Two advantages of using a model?

Cheap and quick, and it lets you predict and test 'what if' changes.

One limitation of using a model?

It simplifies, so predictions may be wrong if assumptions are unrealistic.

Statistical modellingEdexcel International A Level Maths: Flashcards

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What is a mathematical model?

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What is a mathematical model?: A simplified mathematical description of a real-world situation.
Name the steps of the modelling cycle.: Observe, devise a model, use it for predictions, compare with data, refine, report.
What is an assumption in a model?: A simplification that makes the situation easier to analyse.
Three common probability assumptions?: Outcomes equally likely, trials independent, probability constant.
Expected frequency formula?: $np$ for $n$ trials and probability $p$ .
Relative frequency formula?: Observed frequency divided by the total number of trials.
How does the relative frequency behave as the number of trials grows?: It settles close to the model's probability.
What would suggest a model is unsuitable?: Large or systematic differences between observed and expected values.
What would suggest a model is reasonable?: Small, patternless differences that could be due to chance.
Model for a fair 6-sided die?: $P(X=x)=\frac16$ for $x=1,\ldots,6$
Two advantages of using a model?: Cheap and quick, and it lets you predict and test 'what if' changes.
One limitation of using a model?: It simplifies, so predictions may be wrong if assumptions are unrealistic.

Exam questions on Statistical modelling

A game designer models the result $X$ of a single spin of a spinner that has five sectors, numbered 1 to 5.
In 200 spins a 3 occurs 41 times. Comment on whether this result supports the model.2 marks
A bus company states that its 08:15 bus is late with probability 0.15 on any day. A commuter uses this statement as a model for the number of late buses over 20 weekdays.
Find the number of late buses that the model predicts over the 20 days. In fact the bus was late on 9 of the 20 days. Comment on the suitability of the model.2 marks
A teacher models the score $X$ on one roll of a die by $P(X=x)=\frac16$ for $x=1,2,3,4,5,6$ . A student rolls the die 60 times. The scores 1, 2, 3, 4, 5 and 6 occur 7, 12, 9, 11, 8 and 13 times respectively.
(i) State the expected frequency of each score under the model. (ii) Find the relative frequency of the score 6 from the student's data.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).