1. A machine is set to produce bags of sugar whose weights are normally distributed with mean 1000g and standard deviation 5g to check the accuracy. The samples of 9 bags are taken and mean weight is found to be 1003g.

a) Define type I error and type II error

b) Using the one tail test, is the machine's setting set to high?

c) Using a two tail test, is the machine correctly set

2. A random sample of 10 flash  lights batteries with a mean operating life of 58 hours and a sample standard deviation of 1hour is picked from a production line known to be produced battery with normally distributing operating lives. The manufacturer claims that the mean operating life is 8hourd.

a) Define a null hypothesis and an alternative hypothesis

b) Explain the steps in hypothesis testing

c) Test the manufacturers claim at 5%

3. a) State the central limit theorem and discuss the notions of a matched pair comparison test

    b) The distribution of vehicle makes in a certain city in 2010 was as follows:

A recent survey of 700 vehicles from the same city produced the following distribution:

Test at the 90% level of significance whether the distribution of the vehicle makes is significantly different from the 2010 distribution

4.

a) The random variable X has probability function

P(X = x) = 3x -1/20, x = 1,2,3,4.

Construct a table giving the probability distribution of X. Hence or otherwise, find

i. P(2 < X ≤ 5 )

ii. E(X)

iii. Var(X)

iv. var(2-3x)

b) The mean number of defective cheque books in a box is known to be 4. Assuming that the number of defective cheque books follows a Poisson distribution, find the probability that in a box there will be no defective cheque book

5. Packages from  a machine a normally distributed  with a mean 200g and it's standard deviation  2grams. Find the probability that a package from the machine weighs

a) Less than 197g

b) More than 200.5g

c) Between 198.5g and 199.5g

d) State the properties of a normal distribution

Part Two:

1. Suppose that the probability function for the number of errors on a page from a business text book x is given by

Find

a) E[x]

b) Var [x]

c) E[2x² - x +2x]

d) Var [4x +1]

2. A random variable X has an exponential distribution given by

Determine the

a) Value of a

b) Mean of x

c) Variance of x

d) Mean of y = x/2 - 1

3. The joint pdf of random variable X and Y is

Where k is a constant. Find the

a) Value of k

b) fx(x) and fy(y)

c) f(x/y) and f(y/x)

d) Are X and Y independent

4. A particular concentration of chemical found in polluted water has been found to be litho up to 20% of the fish that are exposed for 24 hours. 20 fish are place in a tank containing this concentration of chemical in water. Calculate the probability that

a) Exactly 14 fish survived

b) At least 10 survived

c) At most 4 survived

d) P (2≤ x < 5)

e) mean E[x] and variance[x]

5. A man pays a kwacha a throw to try and win K3 doll for his daughter. His probability of winning on each throw is 0.1. the man is expected to note the geometrical distribution

a) What is the probability that two throws will be required to win the doll

b) What is the probability that more than 3 throws will be required to win the doll

c) What is the expected numbers to win the doll