**UNIVERSITY EXAMINATIONS: 2014/2015**

**ORDINARY EXAMINATION FOR THE BACHELOR OF SCIENCE**

**IN INFORMATION TECHNOLOGY**

**BIT1301 PROBABILITY AND STATISTICS**

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

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

**QUESTION ONE (30 Marks)**

a) The table below shows Marks of students in a mathematics examination.

Marks 12 −14 15 −17 18 −20 21−23 24 −26 27 −29

No. of students 2 6 a 8 4 1

Given that the mean is 19.9 , determine the

i). Value of a ; (5 Marks)

ii). Standard deviation (5 Marks)

b) The length of 64 components produced in a workshop was found to follow a normal

distribution with a mean of 20 mm and standard deviation of 2.5 mm .Calculate the:

i). Probability of lengths less than 19.95 mm ; (3 Marks)

ii). Probability of lengths longer than 22 mm ; (3 Marks)

iii). Number of components having lengths between 18.5 mm and

24.8 mm .

(4 Marks)

c) An exploration firm finds that 5% of the test wells it drills yield a deposit of natural

gas. If the firm drills 6 wells, what is the probability that

i). Exactly 2 wells yield gas.

ii). At least 1 well yields gas.

iii).At most 3 wells yield gas. (6 Marks)

d) The expected number of non-defective bolts in a box is 8 , and the variance is 1.6

.Find the probability that there is only one non-defective bolt in the box.

(4 Marks)

**QUESTION TWO (20 Marks)**

a) The table below shows the average class size and the percentage of students passing

an end term examination in nine schools.

Average class size

X50 60 50 60 80 50 80 40 70

Pass rate Y 30 60 40 50 60 30 70 50 60

i). Find the equation of the least squares line of regression of pass rate on the average

class size. (8 Marks)

ii). Predict the pass rate of the class with an average class of size 64. (2

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 oil checked is 0.29 and 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?

(5 Marks)

c) The Marks of 800 candidates in an examination were normally distributed with

mean of 45 Marks and the standard deviation of 20 Marks. If 5% of the candidates

obtained a distinction by scoring X Marks or more, estimate the value of X.

(5 Marks)

**QUESTION THREE (20 Marks)**

a) State three properties of a normal distribution. (3 Marks)

b) Draw the scatter diagram and calculate the Spearman’s rank correlation coefficient

and interpret it from the following data interpret your results in both cases.

X: 75 32 34 40 45 33 12 30 36 72 41 57

Y: 60 34 40 50 45 41 22 43 42 66 64 46

(10 Marks)

c) The mean and standard deviation of 20 items was found to be 10 and 2 .Later on

it was discovered that the item 12 was misread as 8.Calculate the correct mean

and standard deviation. (7

Marks)

**QUESTION FOUR (20 Marks)**

a) A department supervisor is considering purchasing a photocopy machine. One consideration is how often the machine will need repairs. Let x denote the number of repairs during a year. Based on the past performance, the distribution of x is shown as

follows.

Number of repairs , x

0 1 2 3

P( x) 0.2 0.3 0.4 0.1

i). What is the expected number of repairs during a year? (3

Marks

ii) What is the variance and standard deviation of the number of repairs during a

year? (5 Marks)

b) The percentage of defective components in a manufacturing process of an article is

2%. Find the probability that in a sample of 100 items chosen at random.

i). Exactly 3 are defective (2 Marks)

ii). At least 3 are defective (5 Marks)

c) A particular model of a radio set carries the following price tags(Ksh.)

210 , 210 , 225 , 225 , 235 , 240 , 250 , 270 , 250

Find the mean deviation of the price. (5 Marks)

**QUESTION FIVE (20 MARKS)**

a) Define the following terms as used in probability:

i). Independent event (2 Marks)

ii). Exhaustive event (2 Marks)

b) The domestic electricity consumption by households in a typical urban residence in

Kenya fits a normal distribution with mean of 100kWh and a standard deviation of 16

kWh.

i). Determine the proportion of the households which consume between 85 kWh and

110 kWh. (4 Marks)

ii). The top 20% of the households are considered to be heavy consumers and are

therefore advised to use energy saving devices. Determine the lower limit for the

consumption by the households (4 Marks)

c) From the following data of the heights (in inches) of a group of plants

6.1, 6.2, 6.2, 6.3, 6.1, 6.4, 6.4, 6.0, 6.5, 6.3, 6.4, 6.5, 6.6, 6.4, 6.3

Calculate

i). Mean (2 Marks)

ii). Median (1 mark)

iii).semi interquartile range (2 Marks)

iv). coefficient of skewness (3 Marks)