2022 HSC Mathematics Extension 1
Question 13(e):
Binomial distribution
A chocolate factory sells 150-gram bars. Inspectors randomly select 16 bars and find 8 weigh less than 150 g. The manager claims 80% of bars weigh 150 g or more.
(i) Using the normal approximation to the binomial, calculate the probability of at least 8 underweight bars in a sample of 16, assuming the manager's claim (underweight rate ).
(ii) Explain why the inspectors' normal-approximation method might not be valid.
How to recognise this question
Question type: binomial distribution
- The command and notation point to binomial distribution.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: identify the random variable and its parameters before using the matching probability formula.
Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.
Step-by-step answer
- (i) Let be the number of underweight bars. Need .
- Mean , variance , .
- Without continuity correction: (table) or using the rough split two tails.
- Using proportions: underweight has mean , sd ; .
- NESA (i): 2 marks; 1 for introducing a suitable normal approximation. Accept equivalent stated conventions (with/without continuity correction).
- (ii) The sample size is small; (and ) does not meet the usual rule of thumb and for a good normal approximation to the binomial.
- NESA (ii): 1 mark for a correct validity explanation.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
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.
Question 2
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.
Question 3
A manager claims only of calls go unanswered. In a sample of calls, are unanswered.
(i) Using the normal approximation to the binomial (without continuity correction), calculate for . Hence,
(ii) Explain why the normal-approximation method might not be valid.