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Tue 29 Dec, 2015 12:12 am

(1) In a factory, 10% of products are made by machine A, 30% by B and 60% by C. 8% of

products made by A are defective, whereas the defective rates for B and C are 4% and

1% respectively.

(a) If a product is randomly selected from this factory, what is the probability that

it is made by A and is defective? (2 marks)

(b) If a product is randomly selected from this factory, what is the probability that

it is defective? (2 marks)

(c) If a randomly selected product from this factory is inspected and found to be

defective, find the probability that it is made by (i) A, (ii) B, (iii) C. Verify

that the sum of these 3 answers is 1. (5 marks)

(d) If a product from this factory is not defective, what is the probability that it is

made by A? (3 marks)

(2) A product consists of 4 components: A, B, C, D. The probability that they are defective

is respectively 8%, 6%, 4% and 2%.

(a) The product is considered defective if one or more components are defective.

What is the probability that the product is defective? (4 marks)

(b) If the product is defective, what is the probability that component A is the only

defective component? (3 marks)

(3) 1% of products manufactured by your factory are defective. Suppose 200 units of the

products are randomly selected and tested.

(a) Use Poisson approximation to estimate the probability that at most 2 of the 200

units are defective. (4 marks)

(b) Compare the answer in (a) with the answer obtained without using Poisson

approximation. (4 marks)

(4) In an industrial region, power failures occur randomly and there are on average 3

power failures per month. The numbers of power failures in different months are

independent. Find the probability that in two months, the total number of power

failures is at most 3:

(a) using the reproductive property of the Poisson distribution; (3 marks)

(b) without using the reproductive property of the Poisson distribution. i.e.

consider all the possible combinations of numbers of power failures in the two

months. (4 marks)

(5) The length of short bricks follows a normal distribution with mean 12 cm and standard

deviation 2 mm. The length of long bricks follows a normal distribution with mean

23cm and standard deviation 3 mm.

(a) A short brick and a long brick are chosen randomly. Find the probability that

the length of the long brick is more than two times the length of the short brick.

(4 marks)

(b) A row of bricks consists of 4 short bricks and 3 long bricks arranged end to

end tightly (without any gap in between). Find the probability that the total

length of the row exceeds 118 cm. (4 marks)

(c) Estimate the minimum length of the 10% of longest bricks in a large heap of

long bricks. (3 marks)

(6) The weight of men in a population follows a normal distribution with mean 50kg and

standard deviation 12kg.

(a) Find the probability that the weight of a randomly selected man exceeds 56kg.

(2 marks)

(b) Find the probability that the average weight of 9 randomly selected men

exceeds 56kg. (4 marks)

(c) The maximum capacity of a lift is 1000kg. If a randomly formed group of 19

men enter the lift, find the probability that it is overloaded. (4 marks)

(d) The weight of women follows a normal distribution with mean 45kg and

standard deviation 10kg. Find the probability that the average weight of 9

randomly selected men exceeds the average weight of 4 randomly selected

women. (4 marks)

(7) A random sample of 8 specimens of board made of material 1 are tested for

compressive strength and the results, in 104N/m2, are listed below:

2105 2114 2209 2126 2192 2144 2170 2164

6 specimens of board made of material 2 are tested and the results for compressive

strength in the same units are summarised as Σ(x – 2000) = 192,

Σ(x – 2000)2 = 12944.

Assume that compressive strengths of board of the two materials are normally

distributed with equal variances.

(a) Obtain a 95% confidence interval for the mean compressive strength of

material 1 board. (5 marks)

(b) Obtain a 95% confidence interval for the difference in mean compressive

strengths of boards made of the two materials. (6 marks)

(8) According to past records, 30% of vehicles in an urban area used diesel engines. In a

random sample of 200 vehicles taken from this urban area recently, 73 vehicles are

using diesel engines. In a random sample of 150 vehicles taken from a rural area

recently, 60 vehicles are using diesel engines.

(a) Test at 5% level of significance that the proportion of vehicles in the urban

area using diesel engines has increased. (5 marks)

(b) Test at 5% level of significance that the proportions of vehicles using diesel

engines are the same for the urban area and the rural area in this recent study.

(5 marks)

(9) In a food canning factory, the products are packed in boxes of 24 cans each. If the

gross weight of a can is below 600 g, the can is considered as under-filled. 200 boxes

were randomly selected for checking. The number of under-filled cans in one box is

counted as X. The values of X are distributed as in this table:

X 0 1 2 3 ≥ 4

Number of boxes 67 68 47 18 0

Test, at 5% level of significance, that a Poisson distribution fits the data satisfactorily.

(10 marks)

(10) Three brands of batteries are under study. It is suspected that the lives (in weeks) of

the three brands are different. Four batteries of each brand are tested with the

following results:

Weeks of Life

Brand 1 Brand 2 Brand 3

100 76 108

92 75 96

96 84 98

92 82 100

Test at 5% level of significance whether the mean lives of these brands of batteries are

different. (10 marks)

@deadey3,

pictures of the questions can be referred to if searched in yahoo answers by searching the question "Statistics mathematics help?" my profile name in yahoo is Shen Yi Kuah