2.04f Find normal probabilities: Z transformation

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Pre-U Pre-U 9794/3 2013 November Q6
9 marks Moderate -0.3
A company supplies tubs of coleslaw to a large supermarket chain. According to the labels on the tubs, each tub contains 300 grams of coleslaw. In practice the weights of coleslaw in the tubs are normally distributed with mean 305 grams and standard deviation 6 grams.
  1. Find the proportion of tubs that are underweight, according to the label. [3]
The supermarket chain requires that the proportion of underweight tubs should be reduced to 5%.
  1. If the standard deviation is kept at 6 grams, find the new mean weight needed to achieve the required reduction. [3]
  2. If the mean weight is kept at 305 grams, find the new standard deviation needed to achieve the required reduction. Explain why the company might prefer to adjust the standard deviation rather than the mean. [3]
Pre-U Pre-U 9794/3 2014 June Q6
11 marks Standard +0.3
A machine is being used to manufacture ball bearings. The diameters of the ball bearings are normally distributed with mean 8.3 mm and standard deviation 0.20 mm.
  1. Find the probability that the diameter of a randomly chosen ball bearing lies between 8.1 mm and 8.5 mm. [5]
  2. Following an overhaul of the machine, it is now found that the diameters of 88% of ball bearings are less than 8.5 mm while 10% are less than 8.1 mm. Estimate the new mean and standard deviation of the diameters. [6]
Pre-U Pre-U 9794/3 2014 June Q6
11 marks Standard +0.3
A machine is being used to manufacture ball bearings. The diameters of the ball bearings are normally distributed with mean 8.3 mm and standard deviation 0.20 mm.
  1. Find the probability that the diameter of a randomly chosen ball bearing lies between 8.1 mm and 8.5 mm. [5]
  2. Following an overhaul of the machine, it is now found that the diameters of 88\% of ball bearings are less than 8.5 mm while 10\% are less than 8.1 mm. Estimate the new mean and standard deviation of the diameters. [6]
Pre-U Pre-U 9794/3 2016 June Q2
8 marks Moderate -0.8
The weights of pineapples on sale at a wholesaler are normally distributed with mean \(1.349\) kg and standard deviation \(0.236\) kg. Before going on sale the pineapples are classified as 'Small', 'Medium', 'Large' and 'Extra Large'.
  1. A pineapple is classified as 'Small' if it weighs less than \(1.100\) kg. Find the probability that a randomly chosen pineapple will be classified as 'Small'. [5]
  2. \(10\%\) of pineapples are classified as 'Extra Large'. Find the minimum weight required for a pineapple to be classified as 'Extra Large'. [3]
Pre-U Pre-U 9794/3 2019 Specimen Q1
6 marks Easy -1.2
The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes
  1. more than 27 minutes, [3]
  2. between 20 and 25 minutes. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q1
6 marks Easy -1.2
The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes
  1. more than 27 minutes, [3]
  2. between 20 and 25 minutes. [3]
Pre-U Pre-U 9795/2 Specimen Q7
6 marks Challenging +1.2
The length \(M\) of male snakes of a certain species may be regarded as a normal random variable with mean \(0.45\) metres and standard deviation \(0.06\) metres. The length \(F\) of female snakes of the same species may be regarded as a normal random variable with mean \(0.55\) metres and standard deviation \(0.08\) metres. Assuming that \(M\) and \(F\) are independent, find the probability that a randomly chosen male snake of this species is more than three-quarters of the length of a randomly chosen female snake of this species. [6]
Pre-U Pre-U 9795/2 Specimen Q11
12 marks Standard +0.3
  1. State briefly the conditions under which the binomial distribution \(\text{B}(n, p)\) may be approximated by a normal distribution. [2]
  2. A multiple-choice test has \(50\) questions. Each question has four possible answers. A student passes the test if answering \(36\%\) or more of the questions correctly. Using a suitable distributional approximation, estimate the probability that a student who selects answers to all the questions randomly will pass the test. [5]
  3. A test similar to that in part (ii) has \(N\) questions instead of \(50\) questions. Estimate the least value of \(N\) so that the probability that a student gets \(36\%\) or more of the questions correct, by selecting answers to all questions randomly, will be less than \(0.01\). (A continuity correction is not required in this part of the question.) [5]