RI S3 Normal Distribution Tutorial Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ___________________________ Tutorial S3: Normal Distribution Page 1 of 3 Tutorial S3: Normal Distribution 1 Given that ~ N(1.5, 2.5)X , evaluate (a) P( 0.7)X  , (b) P( 3.7)X  , (c) P(1.1 2.3)X , (d) P( 0.5)X  , (e) P( 0.5)X  (f) P( 1.5 2.5)X  . [(a) 0.306 (b) 0.0821 (c) 0.293 (d) 0.161 (e) 0.839 (f) 0.683] 2 Given that ~ N(22,5)Y and ~N ( 0 , 1 )Z , find the value(s) of k such that (a) P( ) 0.35Yk , (b) P( ) 0.8Yk , (c) P( 25) 0.46kY  , (d) P(| | ) 0.27Zk , (e) P(22 22 ) 0.67kY k   , (f) P( ) 0.3Yk . [(a) 21.1 (b) 20.1 (c) 21.7 (d) 1.10 (e) 2.18 (f) k > 20.8] 3 The random variable X follows a normal distribution with mean  and standard deviation  . Given that P( 20) 0.14X  and that P( 50) 0.65X  , calculate the values of  and  . [ = 42.1,  = 20.5] 4 The independent random variables X and Y are each normally distributed with means 6 and 8 respectively, and variances 9 and 16 respectively. Find (a) (i) P( )XY (ii) P( 1.5)XY . (b) 12, XX and 3X are three independent observations of X and 12 and YY are two independent observations of Y. Find (i) 123P( 15)XXX , (ii) 12 1 2P( )XXY Y  , (iii) 12P(2 )XYY . [(a)(i) 0.655 (ii) 0.218 (b)(i) 0.282 (ii) 0.286 (iii) 0.314] 5 The random variables X and Y are normally distributed with known means and variances. A student was aske d to find the probability that sum of 3 independent observations of X differ from 3 times an observation of Y by at least 1. State an assumption that the student has to make to solve the question. Part of the student’s solution to this question is given as follows:    3 3 ~ N 3E 3E , 9Var 9VarXY X Y X Y  P 3 3 1 ....XY  Point out 3 mistakes that the student has made. [3]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________ Tutorial S3: Normal Distribution Page 2 of 3 6 The pineapples in a large warehouse have mass es with a normal distribution. The mean of this distribution is 1.52 kg and the standard deviation is 0.070 kg. Six randomly chosen pineapples are packed in a box of mass 2.15 kg. Find the probability that the total mass of the filled box is less than 11.2 kg. Find the probability that, out of 3 randomly ch osen filled boxes, at least 1 has total mass less than 11.2 kg. [4] [0.342 , 0.715] 7 A company sells bags of cement in two sizes. The mass of a large bag of cement can be modelled by a normal distribution with mean 50 kg and standard deviation 2 kg. The mass of a small bag of cement can be modelled by a normal distribution with mean 30 kg and standard deviation 1.5 kg. (i) Find the mass that is exceeded by 99% of the large bags. [2] (ii) Find the probability that total mass of three small bags is less than twice the mass of one large bag. [3] (iii) State an assumption needed for your calculation in part (ii). [1] Three large bags are selected at random. (iv) Find the probability that two weigh more than 53 kg and one weighs less than 53 kg. [3] [(i) 45.3 (ii) 0.982 (iv) 0.0125] 8 In this question you should assume that T, W and D follow independent normal distributions. James leaves home to go to work at T minutes past 8 am each day, where T follows the distribution   2N5 , 1 . 2 . (i) Sketch this distribution for the period from 8 am to 8.10 am. [2] (ii) Find the probability that, on a randomly chosen day, James leaves for work later than 8.06 am. [1] When the weather is fine, James walks to work. The time, W minutes, he takes to walk to work follows the distribution   2N 21, 3 . James is supposed to start work at 8.30 am. (iii) Find the probability that, on a randomly chosen day when James walks, he is late for work. [2] When the weather is not fine, James drives to work. He still leaves at T minutes past 8 am each day; the time, D minutes, he takes to drive to work follows the distribution   2N1 9 ,6 . On average, the weather is fine on 70% of mornings. (iv) One day, James is late for work. Find the pr obability that the weather is fine that day. [5] [(ii) 0.202 (iii) 0.108 (iv) 0.606]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________ Tutorial S3: Normal Distribution Page 3 of 3 9 In this question you should state the parame ters of any distributions that you use. A manufacturer produces specialist light bulbs. The masses in grams of one type of light bulb have the normal distribution  2N5 0 , 1 . 5 . (i) Sketch the distribution for masse s between 40 grams and 60 grams. [2] (ii) Two light bulbs are randomly chosen. Fi nd the probability that one of the light bulbs has mass less than 50.4 g and the other has mass more than 50.4 g. [3] Each light bulb is packed into a randomly chosen box. The masses of the empty boxes have the distribution   2N7 5 , 2 . (iii) Find the probability that the total mass of 4 randomly chosen empty boxes is more than 297 grams. [2] (iv) Find the probability that the total mass of a randomly chosen light bulb and a randomly chosen box is between 124.9 and 125.7 grams. [3] In order to protect the bulbs in transit each bulb is surrounded by padding before being packed in a box. The mass of the padding is modelled as 30% of the mass of the bulb. (v) The probability that the total mass of a box containing a bulb and padding is more than k grams is 0.9. Find k. [4] (vi) Find the probability that the total mass of 4 randomly chosen boxes, each containing a bulb and padding, is more than 565 grams. [3] [(ii) 0.478 (iii) 0.773 (iv) 0.126 (v) 137 (vi) 0.163] 10 Every weekday, the last train from Bedok Station leaves the station at 11.29 pm. Its travel time X (in minutes) from Bedok Station to Redbridge Station may be taken to follow a normal distribution with mean 30 and variance 9. (i) Find the travel time exceeded by 90% of the trips made by the last train. [2] At 11.40 pm every weekday, Abel walks from his workplace to Redbridge Station to catch the last train. The time Y (in minutes) he takes follows a normal distribution with mean 16 and variance 4. (ii) Find the probability that the train arrives at Redbridge Station after 12 am and Abel arrives before 12 am. [2] (iii) Find the probability that Abel will not miss the train. [3] (iv) Explain why the answer to part (iii) is greater than the answer to part (ii). [1] The time W (in minutes) taken by Dina to walk from her workplace to Redbridge Station has a mean of 4 and a variance of 6. Explain why W is unlikely to be normally distributed. [1] [(i) 26.2 (ii) 0.361 (iii) 0.797]