Mathematical Methods · Unit 4 · Continuous random variables and the normal distribution · Normal distributions
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A factory produces metal rods whose lengths are normally distributed with a mean of 150 mm and a standard deviation of 3 mm. Rods are classified as acceptable if their length is between 145 mm and 154 mm. Determine the probability that a randomly selected rod is acceptable.
A commercial orchard measures the diameters of apples from two different varieties: Crimson and GoldStar. Both varieties have diameters that are normally distributed. The table below shows summary statistics from this season's harvest. The orchard manager wishes to market apples from both varieties in a premium gift box. To qualify for the gift box, an apple must have a diameter in the largest 20% for its variety. Determine the minimum diameter, correct to the nearest millimetre, for a Crimson apple to qualify for the gift box.
The daily sales (in dollars) of a bakery are normally distributed with a mean of $850 and a standard deviation of $120. Given that P(Z ≤ 1.5) = 0.9332 and P(Z ≤ −0.75) = 0.2266, where Z is a standard normal random variable, determine the probability that the daily sales will be between $760 and $1030.
The daily rainfall at a regional weather station is normally distributed with a mean of 12 mm and a standard deviation of 3.5 mm. Determine the probability that the rainfall on a randomly selected day exceeds 18.5 mm.