Binomial distribution — Smaller $n$, different $p$

Question

A manager claims only of calls go unanswered. In a sample of calls, are unanswered.

(i) Using the normal approximation to the binomial (without continuity correction), calculate for . Hence,

(ii) Explain why the normal-approximation method might not be valid.

This practice question was inspired by Question 13(e) in the 2022 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: identify the random variable and its parameters before using the matching probability formula.(i) , variance , .
Simplify the previous line carefully, keeping exact values where possible..
Simplify the previous line carefully, keeping exact values where possible.(ii) (and is small), so the usual rule of thumb , fails and the normal approximation is not reliable.
Finish the calculation, then check that the result meets the question’s conditions.Using part (i), note that a moderate probability estimate is still undermined by the small-sample rule.
State the final answer clearly in the form the question requested.Answer: (i) about ; (ii) (small ) invalidates the approximation.

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Probability calculation plus a validity critique of the normal–binomial approximation.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.