Binomial distribution — p=0.2, n=800

Question

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

(i) Show that .

(ii) Hence approximate using the standard normal table, 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.Table: .

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

Verification: Hence: use the variance from (i) to standardise in (ii).

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