**UNIVERSITY EXAMINATIONS: 2013/2014**

**ORDINARY EXAMINATION FOR THE BACHELOR OF SCIENCE **

**IN INFORMATION TECHNOLOGY/BUSINESS INFORMATION **

**TECHNOLOGY**

**BIT1301 BUSS 202 PROBABILITY AND STATISTICS**

**DATE: AUGUST, 2014 TIME: 2 HOURS**

**INSTRUCTIONS: Answer Question ONE and any other TWO**

**QUESTION ONE (30 Marks)**

a) The grades of a group of1000students in an exam are normally distributed with amean of70and standard deviation of10. A student from this group is selectedrandomly.

i) Find the probability that his or her grade is greater than

80. (3 Marks)

ii) Find the probability that his or her grade is less than

50. (3 Marks)

iii) Find the probability that his or her grade is between50and80.

(3 Marks)

iv) Approximately ,how many students have grades greater than

80.

(2 Marks)

b) There are24elephants in a game reserve. The warden tags6of the elephantswith small radio transmitters and returns them to the reserve. The next month, herandomly selects5

elephants from the reserve. He counts how many of these

elephants are tagged. Assume that no elephants leave or enter the reserve or die

or give birth, between the tagging and the selection; and that all the outcomes of

the selection are equally likely. Find the probability that exactly two of the

selected elephants are tagged, giving your answer correct to three decimal places.

(4 Marks)

c) Let a random variable

X

have probability density function.

X 0 1 2 3 4

P X x 0.08 0.15 0.45 0.27 0.05

Compute:

i) Expected value of X

ii) Standard deviation of X (6 Marks)

d) Distinguish between binomial distribution and Poisson distribution. (4 Marks)

e) On average one in

200

computers break down in a certain fir per day. Find the

probability that out of a sample of

200

computers selected at random:

i) None breaks down (2 Marks)

ii) More than two breaks breakdown. (3 Marks)

**QUESTION TWO (20 Marks)**

a) In a school,60%of the pupils have access to the internet at home. A group of8students is chosen at random. Find the probability that

i) exactly

5

have access to the internet. (2 Marks)

ii) at least6students have access to the internet. (4 Marks)

b) The following set of data represents the distribution of annual salaries of a random

sample of

100

managers in a large multinational company:

Salary Range ($’000’) Managers

20

but under25 525but under30 10 30but under35 25 35but under40 35 40but under45 25 45but under50 5

i) Calculate the mean and standard deviation. (7 Marks)

ii) Calculate the Karl Pearson’s measure of skewness and comment on it.

(4 Marks)

iii) Calculate

95%confidence level for the annual mean salaries. (3 Marks)

**QUESTION THREE (20 Marks)**

a) Distinguish between correlation and regression. (4 Marks)

b) The Marks scored by students in a mathematics and science tests were as shown in

the table below.

Students

A B C D E F G H I J

Mathematics

89 75 68 74 82 63 66 79 80 58

Science

85 77 74 72 80 68 62 78 83 59

i) Draw a scatter diagram of this data. (3 Marks)

ii) Calculate Pearson’s correlation coefficient correct the answer to four decimal

places. (7 Marks)

iii) Calculate the rank correlation coefficient correct to four decimal places.

(6 Marks)

**QUESTION FOUR (20 Marks)**

a) In a school,

60%of the pupils have access to the internet at home. A group of8

students is chosen at random. Find the probability that

i) Exactly5

have access to the internet. (2 Marks)

ii) At least6

students have access to the internet. (3 Marks)

b) The probability that a driver stopping at a petrol station will have his car tyres

checked is

0.012

, the probability that he will have both oil checked is

0.29and the

probability that he will have both oil and tyres checked is

0.07

.What is the

probability that a driver stopping at this station will have neither his car tyres checked

nor oil? (4 Marks)

c) The Marks of800candidates in an examination were normally distributed with meanof

Marks and the standard deviation of20Marks. If5%of the candidatesobtained a distinction by scoring

X

Marks or more, estimate the value of

X.

(5 Marks)

d) A bag contains6white and9black balls. Two drawings of4

balls are made suchthat (a) the balls are replaced before the second trial (b) the balls are not replaced4

before the second trial. Find the probability that the first drawing will give4black

balls in each case. (6 Marks)

**QUESTION FIVE (20 Marks)**

a) IfP A 0.5, P B 0.3

andP A B 0.2

, obtain the probability that

i)

A occurs but not

B . (3 Marks)

ii) At least one of

A

and

B

occurs. (3 Marks)

iii) Neither of

A

and

B

occurs. (2 Marks)

b) The table below shows test scores and sales for various salespersons. Use it to

answer the questions that follow.

Salesperson 1 2 3 4 5 6 7 8 9 10

Testscore 44 65 50 57 96 94 110 34 79 65

Sales (Ksh ‘000’) 41 60 40 50 80 68 84 70 55 48

i) Using the least square method, determine the regression equation for sales on

testscore. (10 Marks)

ii) Estimate the weekly sales given that a salesperson makes a testscore of 100.

(2 Marks)