Binomial distribution — Different sample and rate

Question

A bakery claims of loaves meet the minimum weight. Inspectors sample loaves and find underweight.

(i) Using the normal approximation to the binomial (without continuity correction), calculate the probability of at least underweight loaves in a sample of , assuming the claim (underweight rate ). 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.

Step by step

Set up the solution: identify the random variable and its parameters before using the matching probability formula.(i) Let . Need .
Simplify the previous line carefully, keeping exact values where possible.Mean , variance , .
Simplify the previous line carefully, keeping exact values where possible..
Simplify the previous line carefully, keeping exact values where possible.(ii) is only on the edge of the usual rule of thumb and ; with the binomial is right-skewed, so the normal approximation may be poor for a far upper tail.
Finish the calculation, then check that the result meets the question’s conditions.Using part (i), interpret the small tail probability against the manager claim.
State the final answer clearly in the form the question requested.Answer: (i) about ; (ii) borderline / skewness makes the approximation unreliable.

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

Verification: Standardise the binomial count, then comment on and skewness.

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

← Back to 2022 HSC Extension 1 paper Question 13(e)