Binomial distribution — p=0.4, n=400

Question

A population has proportion . A random sample of size is taken and is the sample proportion.

(i) Show that .

(ii) Hence use the standard normal distribution to approximate , giving your answer correct to two decimal places.

This practice question was inspired by Question 11(f) in the 2023 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) .
Simplify the previous line carefully, keeping exact values where possible.(ii) , .
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested.From the standard normal table, (two decimal places).

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

Verification: Full (i)–(ii) chain: exact variance fraction, then normal approximation with table.

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