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3. Given that Z is a standard normal variable, by referring to a standard normal distribution table, find thevalues of the following:(a) P ( Z > 1.79)(b) P (1.796 < Z < 2.205)(c) P (1.33 < Z < 0.45)(d) )33.1( Z P

(e) )221.0455.0( Z P

4. Given that Z is a standard normal variable, find the value of k in each of the following:(a) 0853.0)( = k Z P

(b) 7704.0)( = k Z P

(c) 24.0)( = k Z P

(d) 7499.0)( = k Z P

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5. X is a random variable of a normal distribution with a mean of 50 and a standard deviation of 14. Find(a) the z -score if X = 72.4,(b) )64( X P

6. X is random variable of a normal distribution with a mean of 9.5 and a variance of 2.25. Find

(a) the z -score if X = 10,(b) )5.115.9( X P .

7. The diameters of the ball bearing produced by a factory have a normal distribution of mean 1 cm andstandard deviation 0.2 cm. Find the probability that a ball bearing that is chosen randomly has adiameter that is

(a) less than 0.74 cm,(b) between 0.9 cm and 1.3 cm.

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8. The ages of teachers in a rural school are normally distributed with a mean of 38.2 years and a varianceof 16 years 2.Calculate(a) the probability that the age of a teacher, selected at random, lies between 40 and 50 years.(b) the percentage of teachers who are below 36 years.

9. The masses of durians produced by an orchard follow a normal distribution with a mean of 3.6 kg and astandard deviation of 0.4 kg.(a) Calculate the probability that a durian chosen randomly from the orchard has a mass not more

than 4.2 kg.(b) Find the value of k if 65% of the durians have a mass of more than k kg.

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10. The masses of the boys in a school are normally distributed with a mean, and a standard deviation, . Given 5 % of the boys are less than 42 kg and 10 % of the boys are more than 56 kg. Calculate thevalue of and of .

11. In a cross country event participated by 500 students, the time taken for the run follows a normaldistribution with a mean of 20 minutes and a standard deviation of 10 minutes.

Participants will not be given any point if they take more than 32 minutes to complete the run.(a) If a participant is chosen at random, find the probability that the participant does not get any point.

(b) Medals were awarded to the first 80 participants who took less than t minutes to complete therun. Find the value of t . [SPM 2011, P2, 6 marks]

{(a) 0.1151 (b) 10 mins

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Exercise

(1) Given that Z is a standard normal variable, find the values of the following:(a) )472.1( Z P

(b) )557.0( Z P

(c) )88.1( Z P

(d) )2.145.0( Z P

(e) )944.1106.2( Z P

(2) A random variable X has a normal distribution with a mean of 25 and a standard deviation of 1.6. Find the z -score which corresponds to(a) X = 30,(b) X = 22,(c) X = 25.

(3) A random variable X has a normal distribution with a mean of 30 and a variance of 2 . Giventhe standard score for X = 42 is 3. Find(a) the value of .(b) the probability of X having a value of more than 35.

(4) The diagram below shows a standard normal distribution graph.

If 0( P < z < k ) = 0.2873, find the value of z P ( > k )

(5) The mass of a packet of rice sold at a shopping complex follows a normal distribution with amean of 500 g and a standard deviation of 20 g. If a packet is selected at random, calculate the

probability that its mass is between 480 g and 495 g.

(6) The ages, in years, of children who watch a cartoon movie in a cinema are normally distributed

with a mean of 9 and a variance of 16. Find(a) the standard score of the age of 12,(b) the age that corresponds to the standard score of 0.75,(c) the probability that the ages of children who watch the cartoon movie are between 8 and 12.

(7) The mass of a packet of bread produced by a factory is normally distributed with a mean of 420g and a standard deviation of 5 g. Find(a) the probability that the mass of a packet of bread chosen at random will be between 410 and 415

g,(b) the percentage of the packets of breads whose masses exceed 416 g.

(8) The volumes of the cartridges produced by a factory are normally distributed with a mean of 20m and a standard deviation of 2 m . Find, correct to four significant figures,(a) the probability that a cartridge chosen at random has a volume of less than 23m ,(b) the value of v if 70% of the cartridges have a volume of more than v m .

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0 k z

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Ans:

(1) (a) 0.07050 (b) 0.2884 (c) 0.9699 (d) 0.5585 (e) 0.04370

(2) (a) 3.125 (b) 1.875 (c) 0

(3) (a) 4 (b) 0.1056

(4) 0.2127

(5) 0.2426

(6) (a) 0.75 (b) 6 (c) 0.3721

(7) (a) 0.1359 (b) 78.81

(8) (a) 0.9332 (b) v = 18.95

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