QUESTION ONE

(a) A random sample of size 30 is taken from Bin (20, 0.6) i.e. Binomial distribution. Find

i. (< 12.2)

ii. (> 12.4)

iii. (12.2 < < 12.4)

(b) 2% of the trees in a plantation are known to have a certain disease. What is the probability that, in a sample of 250 trees

i. less than 1% are diseased?

ii. more than 4% are diseased?

(c) The weekly wages (to the nearest K) of the production line workers in small factory is

shown in the table below:

Weekly wage (K) Number of workers

16 - 25 5

26 - 35 16

36 - 45 14

46 - 55 22

56 - 65 26

66 - 75 14

i. Calculate the estimate of the mean wage

ii. Find an estimate for the median wage

iii. Find an estimate for the modal wage

iv. Find an estimate for the quartile deviation wage

v. Compute the coefficient of variation

QUESTION TWO

(a) Assume that the probability of an individual coal miner being killed in a mine accident during a year is 1/2400. Use the Poisson approximation to the Binomial to calculate the probability that in a mine employing 200 miners, there will be atleast one fatal accident in a year.

(b) A package of 8 AA batteries contains 2 batteries that are defective. A student randomly select four batteries and replaces the batteries in his calculator.

i. What is the probability that all four batteries work?

ii. What is the mean and variance for the number of batteries that work?

(c) The probability that a driver passes the written test for a driver's license is 0.75. What is the probability that a person will fail the test on the first try and pass the test on the second try?

(d) The principal of a college wants to estimate the proportion of smokers among his students. What size of a sample should be selected so as to have the proportion of smokers not to exceed by 20% with almost certainty? It is believed from previous records that the proportion of smokers was 0.55.

(e) Ten students were given intensive coaching for a month in statistics. The scores obtained in tests 1 and 7 are given below:

Serial No. of students 1 2 3 4 5 6 7 8 9 10

Marks in 1st test 50 52 53 60 65 67 48 69 72 80

Marks in 7th test 65 55 65 65 60 67 49 82 74 86

Does the scores from test 1 to test 7 show an improvement ? Test at the 2% level of

significance.

QUESTION THREE

(a) A bag contains 10 white, 15 red and 8 green balls. A single draw of 3 balls is made.

i. What is the probability that a white, a red and green balls are drawn?

ii. What would be the probability of getting all the three white balls?

(b) A and B are two events, not mutually exclusive, connected with a random experiment E

. If P (A) = p, P (B) = 2p and P( ) = P² , and,p ( = 0.65 find the following:

i. The value of p

ii. P ()

iii. P ()

(c) Three persons A, B and C are being considered for the appointment as Vice Chancellor of Zambian Open University whose chances of being selected for the post are in the proportion 4 : 2 : 3 respectively. The probability that A , if selected will introduce democratization in the university structure is 0.3 ,the corresponding probabilities for B and C doing the same are 0.5 and 0.8 respectively.

i. What is the probability that democratization is introduced in the university?

ii. The democratization is introduced in the university; what is the probability that it was introduced by

(a) A

(b) B

(c) C

QUESTION FOUR

(a) Suppose a company hires both MBA's and non - MBA's for the same kind of managerial task. After a period of employment some of each category are promoted and some are not.

The table below gives the proportion of the company's managers among the said classes:

MBA Non - MBA

Promoted 0.42 0.18

Not promoted 0.28 0.12

If an employee is chosen at random, what is the probability that an employee is

i. MBA graduate?

ii. Promoted?

iii. MBA graduate given that is promoted?

iv. Promoted given that is MBA graduate?

v. Find out whether MBA qualification and promotion are independent events.

Explain your answer.

(b) The research unit in an organization wishes to determine whether scores on scholastic aptitude test are different for male and female applicants. Random samples of applicants file are taken and summarized below:

Applicants

Female Male

502.1 510.5

S 86.2 90.4

n 399 204

Using the above sample data, test the null hypothesis that the average score is the same for the population male and female applicants. Use 5% significance level and assume that the scores are normally distributed in each case.

(c) A manufacturer claimed that at least 95% of the equipment which he supplied to a factory conformed to specifications. An examination of a sample of 200 pieces of equipment revealed that 18 were faulty. Test his claim at a 1% significance level.

QUESTION FIVE

(a) The monthly demand for transistors is known to have the following probability

distribution:

Demand 1 2 3 4 5 6

Probability 0.10 0.15 0.20 0.25 0.18 0.12

i. Determine the expected demand for transistors

ii. Determine the variance demand for transistors

iii. Suppose that the cost (C) of producing transistors is given by the equation,

C = 10, 000 + 500x . Determine the variance cost.

(b) It is known from the past experience that 80% of the students in a school do their homework. Find the probability that during a random check of ten students:

i. all have done their homework

ii. at most two have not done their homework

iii. at least one has not done the homework

iv. Find the expected number of students who do their homework.

v. Find the standard deviation of students who do their homework.

(c) In a main frame computer centre, execution time of programs follows an exponential distribution. The average execution time of the programs is 5 minutes. Find the probability that the execution time of programs is less than 4 minutes.

QUESTION SIX

(a) A certain firm uses a large fleet of delivery vehicles. Their records over a long period of time (during which their fleet size utilization may be assumed to have remained suitably constant) show that the average number of vehicles per day is 3. Estimate the probability on a given day when

i.all their vehicles will be serviceable

ii.ii. more than 2 vehicles will be unserviceable

iii.iii. exactly 4 vehicles will be unserviceable

(b) A population consists of the numbers 1, 3, 5, 7 and 9.

i. List all possible samples of size two which can be drawn from the population without replacement.

ii. Show that the mean of the sampling distribution of the sample means is equal to

the population mean.

iii. Calculate the variance of the sampling distribution of the sample mean and show that it is less than the population variance.

(c) Suppose that is a discrete random variable with probability mass function

( P x) = cx2 , x = 1, 2,3, 4 .

i. Find the value of c

ii. Find ()

iii. Find E[X (X-1)]