Department of Mathematics
Center for Foundation Studies, IIUM
Semester III, 2013/2014
SHF1124 (MATHEMATICS II)
TUTORIAL 6
CHAPTER 6: THE NORMAL DISTRIBUTION
Section in Textbook
Page No
Questions
8.2 The Standard Normal Distribution
360-361
9, 13, 14
8.3 Applications of the Normal Distribution
366-367
5, 8
8.4 Central Limit Theorem
383-384
12, 13
391
7, 11
8.5 The Normal Approximation to the
Binomial and Poisson Distribution
Review Exercises
394-396
5, 6, 7, 8, 10 (a, b, c), 12(c, d), 13
*Required Textbook: - Salina Mohin et al , Mathematics & Statistics for Pre-University,
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 starfruits. Starfruits weighing less than 38g are graded small,
those weighing more than 49g are graded large and the rest are graded medium. The weight
of starfruits sold are normally distributed with mean 42g and standard deviation 4g.
a) Find
i) the proportion of chokonan mangoes graded small.
ii) the proportion of chokonan mangoes graded medium.
iii) the expected number of large mangoes in a carton of 100 randomly selected
chokonan mangoes from the orchard.
iv) the median weight of the mangoes graded large.
b) The probability that the mean weight of a random sample of n chokonan mangoes
exceeding 43g is 0.1. Find the value of n.
3. The duration of telephone calls made by staff of Aman International College (AIC) is
normally distributed with a mean of 6.5 minutes and a standard deviation of 1.2 minutes.
a) What percentage of telephone calls made by staff of AIC last at least 5 minutes?
b) If the first quartile of the duration of telephone calls made by staff of AIC is m minutes,
determine the value of m.
c) The records of 3 telephone calls made by staff of AIC are selected at random. Find the
probability at least one call lasts for less than 5 minutes.
Page 1 of 3
4. 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.
5. 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.
6. 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
i) exactly 50 bacteria on a plate
ii) between 55 and 75 bacteria on a plate.
iii) less than 550 bacteria on 10 plates
b) A plate is rejected if less than 38 bacteria are found. If 1000 such plates are viewed, how
many will be rejected?.
7. 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.
8. A batik painter knows from past experience that when painting batik cloths, paint spots
occur on the cloth at random and at a rate of 3 for every 900 cm2. 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.
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.
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9. Students with good academic achievement, passed the aptitude test and interview are
given government scholarships to pursue their studies. On graduating they are bonded
to work in government agencies. Of these scholarship holders 5% break their bonds
before their term of service expires and 80% join the private sector after their term of
service expires.
a) A sample of 9 government scholarship holders who have just graduated are selected at
random. Find the probability that exactly 2 of them will break their bond before their
term of service expires.
b) The government awarded scholarships to 60 students in a particular year. By using a
suitable approximation, find the probability that at most 4 of these scholarship holders
will break their bond before their term of service expires.
c) This year the government awarded scholarships to 100 students. By using a suitable
approximation, find the probability that more than 75 of these scholarship holders will
join the private sector after their term of service expires.
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:
Section
8.2
8.3
8.4
8.5
Question
9
13
14
5
8
12
13
7
11
Answer
1.96, 0.25
0.3962, 0.9772, 0.0228
0.6826, 39.66
0.6272, 9.17%, 89
3, 0.6, 0.00621
0.9428, 0.9822, 0.0571
0.0228
0.8485, 0.0968
0.8444, 0.2457, 0.6406
Section
Review
Question
5
6
7
8
10
12
13
Extra Questions:
Question
1
2
3
4
5
6
7
8
9
Answer
100.45, 45.45
a) 0.1587, 0.8012 , 4 , 50.2
b) 26
a) 0.8944
b) 5.696 mins
c) 0.2845
1963.10
a) 0.2241
b) 0.2724, 739
a) 0.0233, 0.6883, 0.0197
b) 2
b) 0.2041 (if p=0.42)
c) 0.8237
a) 0.0771, 0.7254
b) 0.1324
a) 0.0629
b) 0.8153
c) 0.8708
Page 3 of 3
c) 0.5675
Answer
0.2119, 0.3064, smaller
A=75, B=74, 62
0.4681, 0.1068, 0.2327
0.9292, 0.1056
0.2389, 20, 0.1112
0.0052, 81.51
0.5015, 27, 0.9049