Binomial distribution — Show sd approximately 6

Question

Let .

(i) Find .

(ii) By finding , show that the standard deviation of is approximately .

(iii) By using a normal approximation, find the approximate probability that is between and .

This practice question was inspired by Question 12(b) in the 2020 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) , so (exactly, hence ).
Finish the calculation, then check that the result meets the question’s conditions.(iii) Using and , the interval is , i.e. .
State the final answer clearly in the form the question requested.Empirical rule / normal approximation: .

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

Verification: Variance simplifies exactly to ; to is one standard deviation either side of the mean.

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