Department of Mathematics
Center for Foundation Studies, IIUM
Semester I, 2013/2014
SHF1124 (MATHEMATICS II)
TUTORIAL 6
CHAPTER 6: THE NORMAL DISTRIBUTION
Page 1 of 10 Page
Section in Textbook Questions
Number
8.2 The Standard Normal Distribution 360-361 13, 14
8.3 Applications of the Normal 366-367 3, 4, 5, 7
Distribution
8.4 Central Limit Theorem 383-384 6, 12
8.5 The Normal Approximation to the 391 5, 7, 8, 11
Binomial and Poisson Distribution
Review Exercises 394-396 6, 8, 10 (a, b, c), 12, 13
*Required Textbook: - Salina Mohin et al , Mathematics & Statistics for Pre-University,
Page 2 of 10 McGraw-Hill Education (Malaysia) Sdn Bhd.(2013)
EXTRA QUESTIONS
1. The weights of mangoes from a certain orchard follows a normal distribution. A quarter of the mangoes weigh less than 70 g and a third weigh more than 120 g. Find the mean weight and standard deviation of the mangoes produced by the orchard.
2. An orchard markets and sells fresh mangoes. Mangoes weighing less than 38 g are graded
small, those weighing more than 49 g are graded large and the rest are graded medium. The
weight of a mango from a particular tree is normally distributed with mean 42 g and
standard deviation 4 g. Find
a) the proportion of mangoes graded small
b) the proportion of mangoes graded medium
Page 3 of 10 c) the median weight of the mangoes graded large.
3. The mass of babies born in a hospital is normally distributed with a mean of 2.8 kg and a
standard deviation of 0.4 kg.
a) Find the probability that a baby born in the hospital has a mass between 2.6 kg and 3.1
kg.
b) Given that the mass of babies in the first quartile is m kg, determine the value of m.
c) If 10 babies who are born in the hospital is picked at random, calculate the probability
that at least one of the babies will have a mass exceeding 3.4 kg.
Page 4 of 10 4. The average number of milligrams (mg) of sodium in a certain brand of low-salt microwave
frozen dinners is 660 mg, and the standard deviation is 35 mg. Assume the variable is
normally distributed.
a) If a single dinner is selected, find the probability that the sodium content will be more
than 670 mg.
b) If a sample of 10 dinners is randomly selected, find the probability that the mean of the
sample will be larger than 670 mg.
Page 5 of 10 c) Why is the probability for part (a) greater than that for part (b)?
5. The average breaking strength of a certain brand of steel cable is 2000 pounds, with a
standard deviation of 100 pounds. A sample of 20 cables is selected and tested. Find the
sample mean that will cut off the upper 95% of all samples of size 20 taken from the
population. Assume the variable is normally distributed.
6. In a medical research, a new drug is being treated against a specific disease on humans. Clinical tests show that the probability a patient with the disease is cured by taking the new drug is 0.75. a) If 10 patients are treated with the new drug, find the probability that at most 6 of them will be cured. b) If 1000 patients are treated with the new drug, find i) the probability that 758 to 778 patients will be cured. ii) the value of n such that the probability that at least n patients will be cured is 0.8.
Page 6 of 10 7. The number of bacteria on a plate viewed under a microscope follows a Poisson
distribution with a mean of 60.
a) Find the probability that there are between 55 and 75 bacteria on a plate.
b) A plate is rejected if less than 38 bacteria are found. If 2000 such plates are viewed, how
many will be rejected?.
8. Tomatoes from a particular nursery are packed in boxes and sent to a market. Assume that the number of bad tomatoes in a box has a Poisson distribution with mean 0.44. a) Find the probability of there being two or more bad tomatoes in a box when it is opened. b) Using a suitable approximation, calculate the probability that in 80 randomly chosen boxes there will be fewer than 20 bad tomatoes in total.
9. Prepaid phone cards produced by a factory are packed in boxes. Each box contains 100 prepaid phone cards. It is known that 3% of the prepaid phone cards produced are defective. a) Show that the probability that a box chosen at random will contain at most 2 defective prepaid phone cards is approximately 0.42. b) If 15 boxes are chosen at random, find the probability that 6 boxes will contain at most 2 defective prepaid phone cards. c) Eighty boxes are chosen at random. Calculate the probability that between 30 and 50, inclusive, boxes will contain at most 2 defective prepaid phone cards.
Page 7 of 10 10. A batik painter knows from past experience that when painting batik cloths, paint spots
2 occur on the cloth at random and at a rate of 3 for every 900 cm . The batik painter wishes
to paint a piece of cloth 35 cm by 30 cm.
a) Find the probability that after painting the cloth, the cloth will contain
i) exactly 6 paint spots,
ii) not more than 4 paint spots.
b) Eight of these cloths are painted consecutively. Find the probability that 3 of them will
each contain exactly 4 paint spots.
Page 8 of 10 c) It is known that the painter will reject a paint job if there are more than 4 paint spots
found on the cloth. Using a suitable approximation, find the probability that out of 40
pieces of cloth, more than 10 pieces will be rejected.
Among the best of you [are they] who have the best character. The Prophet Muhammad (may Allah bless him and grant him peace)
Questions from textbook:
Sectio Question Answer Section Question Answer n 8.2 13 0.3962, 0.9771, 0.0228 8.5 5 0.1964, 0.1985 14 0.6826, 39.66 7 0.8485, 0.0985 8.3 3 0.7745, 60.6%, 0.0228 8 0.9798, 0.6063 4 0.82% 11 0.8444, 0.2457, 0.6406 5 0.6272, 9.17, 89 Review 6 A=75, B=74, 62 7 5, 40, 0.9759 8 0.9279, 0.1056 8.4 6 0.2135, 0.3085 10 0.2389, 20, 0.1112 12 0.9428, 0.9822, 0.0571 12 0.9387, 0.0052, 81.51
Page 9 of 10 13 0.5015, 27, 0.9049
Extra Questions:
Question Answer 1 100.18, 45.04 2 a) 0.1587 b) 0.8012 c) 50.2 3 a) 0.4649 b) 2.532 c) 0.4991 4 a) 0.3859 b) 0.1841 5 1963.10 6 a) 0.2241 b) i) 0.2724 ii) 739 7 a) 0.6883 b) 4 8 a) 0.0726 b)0.004 9 b) 0.2041 c) 0.8237 10 a) i) 0.0771 ii) 0.7254 b) 0.1324 c) 0.5675
Page 10 of 10