SACE Stage 2 Maths Methods Topic Tests
Normal Distributions Topic Test 1
Number of marks: 9
Writing time: 12 minutes
Instructions
• Show appropriate working and steps of logic in the question booklets
• State all answers correct to three significant figures, unless otherwise instructed
• Use black or blue pen
• You may use a sharp dark pencil for diagrams and graphical representations
The fat content in grams, \(X\), per carton of Ted's Oat Milk can be modelled using a normal distribution with a mean of \(\mu_X = 2.7\) g and a standard deviation of \(\sigma_X = 0.35\) g.
(a) Determine the probability that a carton of Ted's Oat Milk will contain:
(i) at least 2.8 g of fat. (1 mark)
(ii) between 1 g and 3 g of fat. (1 mark)
(b) Ted's Oat Milk is sold in packs of six cartons. Let \(S_6\) represent the distribution of fat content in one six-pack of Ted's Oat Milk.
(i) Explain why \(S_6\) can be modelled by a normal distribution. (1 mark)
(ii) Show that the distribution of \(S_6\) has a mean of \(\mu_{S_6} = 16.2\) g and a standard deviation of \(\sigma_{S_6} = 0.857\) g (correct to three significant figures). (2 marks)
(iii) Hence, determine the probability that a six-pack of Ted's Oat Milk will contain less than 18 g of fat. (1 mark)
(c) Find the value of \(k\) if \(\Pr(Z < k) = 0.99\), given that \(Z \sim N(0,1)\). (1 mark)
(d) The label on the side of a six-pack of Ted's Oat Milk states that: "Each pack of 6 contains less than 18 g of fat."
The company would like this statement to be true for at least 99% of Ted's Oat Milk six-packs and it is considering improvements to the manufacturing process. Although it cannot change the mean fat content of a six-pack through these improvements, the standard deviation can be reduced.
Using your answer to part (c), determine the largest possible value of \(\sigma_{S_6}\) that will result in at least 99% of six-packs containing less than 18 g of fat. Assume that \(\mu_{S_6} = 16.2\) g.
end of booklet
