Binomial distribution — New $n$ and threshold

Question

A plant claims of items pass inspection. In a sample of items, fail.

(i) Using the normal approximation to the binomial (without continuity correction), calculate where is the number of failures under the claim. 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) is only borderline for the usual , criterion, and leaves the distribution somewhat skewed, so the normal tail probability may be inaccurate.
Finish the calculation, then check that the result meets the question’s conditions.Using part (i), relate the calculated probability to the inspectors' observation.
State the final answer clearly in the form the question requested.Answer: (i) about ; (ii) borderline and residual skewness undermine the approximation.

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

Verification: Same two-part structure: approximate probability, then validity.

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