Give a reason why trials might not be independent.

Family or friends behave alike; queues; sampling without replacement from a small population.

Give a reason why p might not be constant.

Different people or times have different probabilities, e.g. pupils living at different distances from school.

When is sampling without replacement still well modelled by a binomial?

When the population is very large, so removing items hardly changes the proportions.

What shape does a Normal distribution have?

Symmetrical, bell-shaped and unimodal, centred on the mean.

What does mean greater than median suggest?

Positive skew, so a symmetrical Normal model may not be appropriate.

Why can a Normal model be unsuitable for a quantity that cannot be negative?

A Normal model gives some probability to negative values, which are impossible.

How do you find the predicted frequency from a model?

Multiply the sample size by the model probability, n\times P(event).

If observed frequency is much higher than model prediction in the upper tail, what does that suggest?

The model underestimates large values; the data may be positively skewed.

Which model suits a continuous measurement and which a count of successes?

Normal for a continuous measurement; binomial for a count of successes.

How should you write a criticism of a model?

Name the condition or feature that fails, explain it in context, and say how it affects the results.

Choosing and critiquing statistical modelsAQA A-Level Maths: Flashcards

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State the four conditions for a binomial model.

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State the four conditions for a binomial model.: Fixed number of trials; two outcomes; independent trials; constant probability of success.
What are the mean and variance of $B(n,p)$ ?: Mean $np$ and variance $np(1-p)$ .
Give a reason why trials might not be independent.: Family or friends behave alike; queues; sampling without replacement from a small population.
Give a reason why $p$ might not be constant.: Different people or times have different probabilities, e.g. pupils living at different distances from school.
When is sampling without replacement still well modelled by a binomial?: When the population is very large, so removing items hardly changes the proportions.
What shape does a Normal distribution have?: Symmetrical, bell-shaped and unimodal, centred on the mean.
What does mean greater than median suggest?: Positive skew, so a symmetrical Normal model may not be appropriate.
Why can a Normal model be unsuitable for a quantity that cannot be negative?: A Normal model gives some probability to negative values, which are impossible.
How do you find the predicted frequency from a model?: Multiply the sample size by the model probability, $n\times P(\text{event})$ .
If observed frequency is much higher than model prediction in the upper tail, what does that suggest?: The model underestimates large values; the data may be positively skewed.
Which model suits a continuous measurement and which a count of successes?: Normal for a continuous measurement; binomial for a count of successes.
How should you write a criticism of a model?: Name the condition or feature that fails, explain it in context, and say how it affects the results.

Exam questions on Choosing and critiquing statistical models

A teacher models the number $X$ of pupils in a class of $30$ who walk to school by the binomial distribution $X\sim B(30,0.4)$ .
The pupils in the class live at very different distances from the school. Explain why $B(30,0.4)$ may not be a suitable model.2 marks
A researcher records the monthly household income $X$ pounds in a city. The sample mean is $2400$ , the sample median is $1900$ and the sample standard deviation is $1800$ . She models $X$ by the Normal distribution $N(2400,1800^2)$ .
State, with a reason, whether the Normal model is appropriate for monthly household income.2 marks
A call centre claims that $70\%$ of calls are answered within $20$ seconds. A manager records the next $10$ calls received on a Monday morning and models the number $X$ answered within $20$ seconds by $X\sim B(10,0.7)$ .
State the distribution of $X$ and find the probability that at least $8$ of the $10$ calls are answered within $20$ seconds.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).