RI S3 Normal Distribution Add Prac Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 1 of 5 Additional Practice Questions for Chapter S3: Normal Distribution 1 A manufacturer produces packets of margarine. The mass of margarine in a packet has a normal distribution with mean 502g and standard deviation 0.8g. Find the proportion of packets which contain less than 500g of margarine. [2] The manufacturer increases the mean am ount of margarine in a packet to  g. The standard deviation remains unchanged. Only 1 packet in 1000, on average, now contains less than 500g. Find  , correct to 1 decimal place. [3] 2 The length of time which an ordinary light-bulb will last may be taken to have a normal distribution with mean 600 hours and standard deviation 100 hours. The length of time for which a new ‘long life’ bulb will last may be taken to have a normal distribution with mean 2000 hours and standard deviation 200 hours. (i) One ordinary bulb is chosen at random. Fi nd the probability that it will last for more than 450 hours. [2] (ii) Two ordinary bulbs are chosen at random . Find the probability that the sum of the times for which they last will be less than 1100 hours. [3] (iii) One ordinary bulb and one long-life bu lb are chosen at random. Find the probability that the long-life bulb lasts for more than three times as long as the ordinary bulb. [3] The total time in which 10 ‘long life’ bulbs will last is denoted by W hours. Find the value of t (to the nearest hour) such that P( ) 0.75Wt . [3] 3 [The notation 2N( , ) denotes a normal distribution with mean  and variance 2 .] The school bus leaves the stop near Ben’s home at X minutes past 8.00 a.m., where X is an 2N(20,3 ) random variable. Ben reaches the bus stop at Y minutes after 8.00 a.m., where Y is an 2N(15,2 ) random variable, X and Y being independent. Find the probability (to three decimal places) that Ben misses the bus. [2] The ride from this stop to the school lasts T minutes, where T is an 2(30,( 7) )N random variable, independent of X. The random variable W is the number of minutes before 9.00 a.m. at which the bus arrives at the school. Express W in terms of X and T. [1] Calculate the mean and variance of W and find, to three significant figures, the probability that the bus arrives at the school after 9.00 a.m. [2]

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 2 of 5 4 Chickens and turkeys are sold by weight. Th e masses, in kg, of chickens and turkeys are modeled as having independent normal distributions with means and standard deviations as shown in the table. Chickens are sold at $3 per kg and turkeys at $5 per kg. (i) Find the probability that a randomly chosen chicken has a selling price exceeding $7. [2] (ii) Find the probability of the event that both a randomly chosen chicken has a selling price exceeding $7 and a random ly chosen turkey has a selling price exceeding $55. [3] (iii) Find the probability that the total selling price of a randomly chosen chicken and a randomly chosen turkey is more than $62. [4] (iv) Explain why the answer to part (iii) is greater than the answer to part (ii). [1] 5 The speed of a randomly chosen passenger car travelling from point A to point B on an expressway has a normal distribution with mean 85 km/h and standard deviation 20 km/h. It is assumed that a passenger car travels at a constant speed from point A to point B. (i) For passenger cars travelling from point A to point B, find the probability that the total speed of 3 randomly chosen cars differs from twice the speed of a randomly chosen car by at most 50 km/h. [3] (ii) The speeds of 80 randomly chosen passenger cars which travelled from point A to point B are recorded. Find the probability that the total distance travelled in 5 minutes is at most 550 km. State an assumption you had made in your calculation. [4] 6 A small component for a machine is made from two metal spheres joined by a short metal bar. The masses in grams of the spheres have the distribution 2N(20,0.5 ). (i) Find the probability that the mass of a randomly selected sphere is more than 20.2 grams. [1] In order to protect them from rusting, the spheres are given a coating which increases the mass of each sphere by 10%. (ii) Find the probability that the mass of a coated sphere is between 21.5 and 22.45 grams. State the distribution you use and its parameters. [3] (iii) The masses of the metal bars are norma lly distributed such that 60% of them have a mass greater than 12.2 grams and 25% of them have a mass less than 12 grams. Find the mean and standard deviation of the masses of metal bars. [4] (iv) The probability that the total mass of a component, consisting of two randomly chosen coated spheres and one random ly chosen bar, is more than k grams is 0.75. Find k, stating the parameters of any distribution you use. [4] Mean mass Standard deviatio n Chickens 2.2 0.5 Turkeys 10.5 2.1

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 3 of 5 7 In this question you should state the parameters of any normal distributions you use. A company makes 3-legged wooden stools fro m 4 solid components - a seat in the form of a disc, and 3 legs each in the form of a long, thin cylinder. The seats and legs are bought in bulk from another company. Over a period of time it is found that the masses of the seats are normally distributed; 80% of the seats have mass less than 2.1 kg, and 15% of the seats have mass less than 1.95 kg. (a) Find the mean mass of the seats and show that the standard deviation is 0.0799 kg, correct to 3 significant figures. [3] The masses of the legs, in kg, follow the distribution N(1.2, 0.02 2). (b) Find the expected number of legs with mass more than 1.21 kg in a randomly chosen batch of 500 legs. [2] (c) Find the probability that the total mass of a randomly chosen seat and 3 randomly chosen legs is between 5.6kg and 5.7kg. [3] In order to make the stools, circular holes are drilled in the seats and the legs are fitted into them. In this process, the mass of seats is modelled as being reduced by 9% and the masses of the legs are unchanged. (d) Find the probability that the total mass of a randomly chosen drilled seat and 3 randomly chosen legs is less than 5.6 kg. [3] The holes made in the seats have di ameters, in mm, that follow the distribution N(31, 0.4 2) and the diameters of the legs, in mm , follow the distribution N(30.7, 0.32). If the diameter of a leg is greater than the diameter of a hole, then the leg has to be sanded down to make it fit. If the diameter of a hole is more than 0.8 mm greater than the diameter of a leg, then padding has to be added when the leg is glued to the seat. (e) A stool is made of a randomly chosen drilled seat and 3 randomly chosen legs. The legs are paired up with the holes at random. Find the probability, that the 3 legs can be fitted without the need for any sanding or padding. [4]

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 4 of 5 8 A group