SAJC Check your understanding Normal Distribution Teacher

Check your Understanding (Normal Distribution) Section 1: Probability 1. Drill & Practice Given that   2~ N 120,2X . Find (i)  P 119X  (ii)  P 125X  (iii) P(118 121)X Answers (i) 0.691 (ii) 0.215 (iii) 0.533 (i)  P 119 0.691X  ] (ii)  P 125 0.994X  (iii) P(118 121) 0.533X   2. YJC Promo 8865/2018/Q7 In a national park, it is known that the lengths of garden snails are normally distributed with mean 3.42 cm and standard deviation 1.10 cm. It is found that 70% of them measure less than 4.0 cm and 80% measure at least 2.5 cm. (ii) Find the probability that the length of a randomly selected garden snail is within ± 0.3 cm of the mean length. [2] Answers (ii) 0.215 YJC Promo 8865/2018/Q7 (ii)     2 2 ~ N 3.42, 1.10 or ~ N 3.42417, 1.09808 G G  P 0.3 0.3 G      3.42 0.3 3.42 0.3 0.214937 0.215 (to 3 s.f.) G       [or 0.21530] 3. JPJC JC1 Promo 8865/2019/Q7 A butcher sells chicken in two different cuts, leg and wing. The masses, in kilograms, of these two cuts have independent normal distributions. The means and standard deviations of these distributions, and the selling prices, in $ per kilogram, are shown in the following table. Mean Standard deviation Selling price Leg 0.35 0.010 12 Wing 0.09 0.003 9 Stating clearly the mean and variance of all distributions that you use, find the probability that

(i) the mass of a randomly chosen wing is within 0.005 kg of the mean mass of wings, [2] (ii) out of 3 randomly chosen wings, exactly one is less than 0.088 kg, exactly one is more than 0.09 kg and exactly one is between 0.088 kg and 0.09 kg, [3] JPJC JC1 Promo 8865/2019/Q7 (Solutions) Let L and W denote the mass of a leg and wing respectively. 2 2 ~ (0.35,0.010 ) ~ (0.09,0.003 ) LN WN (i) P(0.085 0.095) 0.90442 0.904W    (ii) Required probability = P( 0.088) P( 0.09) P(0.088 0.09) 3! (0.25249) (0.5) (0.24751) 3! 0.18748 0.187 W W W            4. MI Prelim 8865/2018/Q9 In this question you should state clearly the values of the parame ters of any normal distribution you use. In a particular stall, the masses, in kilograms, of a particular type of fish called groupers have a mean mass of 0.5 kg and standard deviation 0.0182 kg. (ii) Find the probability that the mass of a randomly chosen grouper is within 0.01 kg of the mean mass of groupers in the stall. [2] Answers (ii) 0.417 MI Prelim 8865/2018/Q9 (ii) Let G be the random variable denoting the mass, in grams, of a randomly selected groupers.   2~ N 0.5, 0.0182G    P 0.5 0.01 0.5 0.01 P 0.49 0.51 0.417 (3 sf) GG        5. PJC Prelim 8865/2018/Q12 The masses of oranges sold by a supermarket have a normal distribution. The mean and standard deviation of the distribution is 0.175 kg and 0.09 kg respectively. Find the probability that an orange chosen at random has mass (i) at most 0.12 kg, [1] 0.5 0.5+0.01 0.5 –0.01 G

(ii) within 0.05 kg of the mean. [2] Answers (i) 0.271 (ii) 0.421 PJC Prelim 8865/2018/Q12 Let R be the random variable denoting the mass, in grams, of a randomly selected orange.   2~ N 0.175,0.09R (ii)  P 0.12 0.27056 0.271R   (ii)  P 0.05 0.175 0.05R     P 0.125 0.225 0.42149 0.421 R    6. CJC Prelim 8865/2018/Q9(i) Kickers chocolates are sold in tins of 5 chocolates. The masses, in grams, of the individual Kickers chocolates and the empty tins have independent normal distributions with means and standard deviations as shown in the following table. Mean Standard Deviation Individual Kickers Chocolate 53 2.8 Empty Tin 15 0.4 (i) Find the probability that two randomly chosen Kickers chocolate each weigh more than 50 grams. [1] Answers (i) 0.736 CJC Prelim 8865/2018/Q9 (i) Let K be the random variable denoting the mass, in grams, of a randomly selected Kickers chocolate.   2~ N 53, 2.8K       12P 50 P 50 0.7361838758 0.736 to 3 s.f. KK     7. HCI Prelim 8865/2018/Q12 A supermarket sells two types of strawberries, A and B. Each type of strawberry comes in different sized packaging. The masses, in kg, of a packet of type A strawberries and a packet of type B strawberries are modelled as having independent normal distributions with means and standard deviations as shown in the table. Strawberries Mean Standard deviation Type A 1.2 0.2 Type B 1.1 0.1 Type A strawberries are sold at $20 per kg and type B strawberries at $26 per kg.

Audrey picks one packet of type A strawberries and one packet of type B strawberries. (i) Find the probability that none of the two packets picked by Audrey weighs less than 1.2 kg. [2] Answers (i) 0.0793 HCI Prelim 8865/2018/Q12 Let X and Y be the weight (in kg) of a packet of type A and type B strawberries respectively.     2 2 N 1.2,0.2 N 1.1,0.1 X Y    P 1.2 P 1.2 0.0793XY   8. NJC Prelim 8865/2018/Q11 In this question you should state clearly the values of the parameters of any normal distribution you use. A supermarket sells two types of durians, Red Prawn and Black Gold. The masses, in kilograms, of the durians each have independent normal distributions. The means and standard deviat ions of these distributions, and the selling prices, in $ per kilogram, are shown in the following table. Mean mass (kg) Standard deviation (kg) Selling price ($ per kg) Red Prawn 0.25 0.02 1.50 Black Gold 0.35 0.03 2.40 (ii) Three Red Prawn durians are randomly selected. Find the probability that exactly one of the durians has mass less than 0.24 kg and exactly one of the durians has mass more than 0.26 kg. [2] Answers (ii) 0.219 NJC Prelim 8865/2018/Q11 Let A be the random variable denoting the mass of a Red Prawn durian.   2N 0.25,0.02A (ii)      P 0.24 0.2P 0.24 P 0. 6 3! 0.219 26A A A      Section 2: Use of Inverse Normal 1. Drill & Practice Given that   2~ N 120,2X . Find (i)  P 0.5Xx (ii)  P 0.5Xx

(iii)  P 0.7Xx (iv) 12P( ) 0.77x X x   , such that 1x and 2x are symmetrical about the mean. (v) P( 120) 0.2xX   Answers (i) 120 (ii) 120 (iii) 119 (iv) 117, 122 (v) 119 (i)  P 0.5Xx Using GC, 120x (ii)  P 0.5Xx Using GC, 120x (iii)  P 0.7Xx Using GC, 119x (iv) 12P( ) 0.77x X x   Using GC, 12 117, 122xx (v)         P( 120) 0.2 P 120 P 0.2 P P 120 0.2 0.5 0.2 0.3 xX X X x X x X               Using GC, 119x 2. AJC Prelim 8865/2018/Q12(i) The weights of the lemons sold on the market stall are normally distributed with mean weight 0.1 kg and standard deviation 0.05 kg. (i) Find the value that is exceeded by 75% of the weights of the lemons. [1] Answers 0.0663a AJC Prelim 8865/2018/Q12 Let L be the random variable denoting the weight, in kilograms, of a randomly selected lemon.   2~ N 0.1, 0.05L  P 0.75 0.0663L a a    3. CJC Prelim 8865/2018/Q9(iii) The masses, in grams, of the individual Venus chocolates have a normal distribution with mean 35 grams and standard deviation  grams. It is given that 85% of Venus chocolates weigh more than 34 grams. (iii) Find  , giving your answer correct to 4 decimal places. [3] Answers (iii) 0.9648 CJC Prelim 8865/2018/Q9 (iii) Let V be the random variable denoting the mass, in grams, of a randomly selected Venus chocolate.

  2~ N 35, V      P 34 0.85 P 34 0.15 34 35P 0.15 1P 0.15 V V Z Z     