# Probability and Statistics

a) If the probability of being a smoker among a group of cases with lung cancer is .6, whats the probability

that in a group of 8 cases you have;

i less than 2 smokers

ii More than 5

iii What are the expected value and variance of the number of smokers

b) The manufacturer of the disk drives in one of the well-known brands of microcomputers expects

2% of the disk drives to malfunction during the microcomputers warranty 17 period. Calculate the

probability that in a sample of 100 disk drives, that not more than three will malfunction

c) At a parking place the average number of car-arrivals during a specified period of 15 minutes is 2.

If the arrival process is well described by a Poisson process, find the probability that during a given

period of 15 minutes

i) no car will arrive

ii) atleast two cars will arrive

iii) atmost three cars will arrive

iv) between 1 and 3 cars will arrive

d) Explain what do you understand by random experiment and a random variable. Briefly explain the

following:

i) Discrete and continuous random variables

ii) Discrete probability distribution.

e) The weekly wage of 2000 workmen is normally distribution with mean wage of \$ 70 and wage standard

deviation of \$ 5. Estimate the number of workers whose weekly wages are

i) between \$ 70 and \$ 71

ii) between \$ 69 and \$ 73

iii) more than \$ 72

iv less than \$ 65

1

STA 2200 W1-2-60-1-6

f) A random sample of 64 sales invoices was taken from a large population of sales invoices. The average

value was found to be \$.2000 with a standard deviation of \$.540. Find a 90 per cent confidence interval

for the true mean value of all the sales.

g) A manufacturer of a new motorcycle claims for it an average mileage of 60 km/liter under city condi-

tions. However, the average mileage in 16 trials is found to be 57 km, with a standard deviation of 2

km. Is the manufacturers claim justified?

h) The table below gives a probability distribution of a discrete random variable X. Given that

P (X < 13) = 0.75, find the value of k and q hence calculate E(X) and V ar(X)

X 4 8 12 15 20

P(X=x) k 0.25 0.3 q 0.1

i) Let X be a continuous random variable. Show that the function

f(x) =

12x , 0 ≤ x ≤ 20 , otherwise i) P (0 ≤ x ≤ 1)

ii P (−1 ≤ x ≤ 1)

iii) E(X) and Var(X) [10 marks]