Calculate probabilities and quantiles associated with a given normal distribution, using technology.

A quality control inspector examines the length of steel rods produced by a manufacturing process. The lengths are normally distributed with a mean of \(125.0\) cm and a standard deviation of \(2.4\) cm. (a) Calculate the probability that a randomly selected rod has a length between \(122.0\) cm and \(128.5\) cm. (2 marks) (b) The inspector rejects rods that fall outside the middle \(95\%\) of the distribution. Calculate the minimum and maximum acceptable lengths for rods to pass inspection. (3 marks)

The diameter D (in centimetres) of mature trees in a managed plantation is normally distributed with mean μ = 42.8 cm and standard deviation σ = 6.5 cm. Refer to the diameter classification table below. (a) Calculate the probability that a randomly selected mature tree has a diameter in Class C. (2 marks) (b) Calculate the probability that a randomly selected mature tree is in Class A or Class E. (2 marks) (c) Determine the diameter that marks the boundary for the top 12% of all mature trees, correct to one decimal place. (1 mark)

The mass of apples sold at a market stall is normally distributed with a mean of 150 g and a standard deviation of 12 g. (a) Calculate the probability that a randomly selected apple has a mass between 130 g and 170 g. (2 marks) (b) The stall owner wants to label the heaviest 10% of apples as 'premium'. Calculate the minimum mass required for an apple to be classified as premium. (3 marks)