Statistical models and their assumptions

What you’ll learn

• Translate a situation into a probability model.
• Explain assumptions and use observations to assess a model.

Understand the idea

A statistical model is a deliberately simplified description of a source of variation. Start by identifying the outcome, the possible values and the conditions of observation. A listed set of mutually exclusive, exhaustive outcomes has probabilities totalling one. A model for a randomly selected day may fail if the data were collected only at weekends.

Probabilities describe the model, while relative frequencies describe observations. Larger samples usually make relative frequencies more stable, but a large biased sample can still misrepresent its population. A discrepancy prompts investigation of the assumptions; it is not automatically proof that a model is wrong.

State assumptions in the context of the question. Random mixing can support equal selection probabilities, replacement can keep probabilities constant, and independent trials mean that knowing earlier results does not alter the next probability. Fit the model, compare predictions with data, and revise it when the evidence warrants a change.

Key formulae

$$0\leq P(A)\leq1$$

$$\sum_{i=1}^{k}P(A_i)=1$$

$$\text{relative frequency}=\frac{\text{number of occurrences}}{\text{number of trials}}$$

Worked example

A bag contains three red and two blue counters. A counter is drawn, replaced and the bag mixed, twice. Model the probability of two red counters and state the assumptions.

1. With thorough mixing and equally selectable counters, $P(R)=3/5$.
2. Replacement restores the original composition. Assume the second draw is independent of the first.
3. Multiply the two red probabilities: $P(RR)=(3/5)^2=9/25$.

Answer: $P(RR)=0.36$, assuming random selection and independent draws after replacement.