Choosing and critiquing statistical modelsAQA A-Level Maths: Revision notes

A binomial model $X\sim B(n,p)$ counts the number of successes in $n$ trials, with $P(X=r)=\binom nr p^r(1-p)^{n-r}$. It is only appropriate when all of these hold:

• the number of trials $n$ is fixed;
• each trial has two outcomes, success or failure;
• the trials are independent;
• the probability of success $p$ is constant for every trial. The mean is $np$ and the variance is $np(1-p)$. In an answer, always say which condition holds or fails, and apply it to the context given.

Look for a reason in the context for each condition to fail:

• Not independent: members of a family or friendship group who behave alike; calls or customers in a queue; selecting from a small population without replacement, since each removal changes the proportions left.
• Probability not constant: pupils at different distances from school; a busy period compared with a quiet one; people with different abilities.
• Not a fixed number of trials, or more than two outcomes with no way to group them into success and failure. Sampling without replacement from a large population changes the probabilities very little, so the binomial model is then a good approximation. For example, from a tank of $50$ fish with $10$ undersized, $P(\text{first two not undersized})=\frac{40}{50}\times\frac{39}{49}=0.637$, close to $0.8^2=0.64$.

The Normal model suits a continuous quantity with a single peak and a roughly symmetrical, bell-shaped distribution. Useful checks:

• the mean and median are close; a mean much greater than the median suggests positive skew, and a mean much less suggests negative skew;
• almost all values lie within about $3$ standard deviations of the mean;
• the model should not give noticeable probability to impossible values, for example negative times, masses or incomes;
• the shape of a histogram is roughly bell-shaped.

Use the model to predict a frequency and compare it with what was observed. Expected number $=n\times P(\text{event})$. Example: the model $L\sim N(31,4^2)$ for fish lengths gives $P(L>38)=P(Z>1.75)=0.0401$. In a sample of $200$ fish, the model predicts $200\times0.0401=8.0$ fish longer than $38$ cm. If $27$ are observed, this is more than three times the prediction, so the model underestimates the upper tail. That suggests positive skew, and the Normal model is not suitable. Similarly, if a Normal model predicts $9\%$ negative incomes for a quantity that cannot be negative, the model is weak in the lower tail.

• A count of successes in a fixed number of independent trials with the same probability: binomial.
• A continuous measurement (time, mass, length) with a symmetrical bell-shaped pattern: Normal.
• A binomial with large $n$ and $p$ not close to $0$ or $1$ can be approximated by $N(np,np(1-p))$. A good answer gives the model, checks its conditions against the context, comments on any that fail, and says what effect this has on the results. For example, if a busy morning lowers the true probability below $0.7$, the binomial model overestimates the chance of answering quickly.