# Probability And Statistics I Question Paper

Probability And Statistics I

Course:

Institution: Kenyatta University question papers

Exam Year:2010

KENYATTA UNIVERSITY

UNIVERSITY EXAMINATIONS 2009/2010

INSTITUTIONAL BASED PROGRAMME (IBP)

EXAMINATION FOR THE DEGREE BACHELOR OF SCIENCE

SMA 160: PROBABILITY AND STATISTICS I

DATE:

Thursday 29th April 2010

TIME: 11.00a.m – 1.00p.m

INSTRUCTIONS: Answer Question one and any other two questions

Q1.

a)

Differentiate between the following terms

i)

Independent and dependent events

ii)

Mutually exclusive and mutually exhaustive events

iii)

Primary and Secondary sources of data

iv)

Continuous

and

discrete

variables

[8marks]

b)

A bag contains 5 red balls, 3 white balls and 7 blue balls. What is the

probability that a ball drawn at random will be,

i)

White

[1mark]

ii)

White or red

[2marks]

iii)

Neither

white

nor

blue [3marks]

c)

An incomplete distribution is given below

Variable 0-10 10-20 20-30 30-40 40-50

Frequency 14

f1 27 f2 15

If the median is 25 and the mode is 24 obtain the missing frequencies f1

and

f2.

Page 1 of 4

1

1

1

d)

Suppose

P( )

A = , P(B) =

and P(A n C) =

where A and C are

2

3

12

independent.

1

If P(B ? C) = show that B and C are mutually exclusive. [4marks]

2

e)

The first four moments of a distribution about the value 4 of a variable are

-1.5, 17,-30 and 108. Obtain the mean variance of the distribution.

[4marks]

f)

Given that the set (a, b, c ,d, e) has mean m and standard deviation s, write

down the mean and standard deviation of the set (a+k, b+k, d+k, e+k) in

terms of m and s respectively where k is a constant.

[2marks]

Q2.

Consider the following frequency distribution table.

Class

0-10

10-20 20-30 30-40 40-50 50-60

Interval

Frequency

15

17 19 27 19 12

a)

Find the mean and the standard deviation by changing both the location

(use an assumed mean of 35) and the scale (use common class width).

[7marks]

b)

Find

i)

Mode

ii)

Median

iii)

Interquartile

range

iv)

Coefficient

variation

v)

Coefficient

of

skewness

[13marks]

Q3.

a)

Out of 100 people, 25 are NARC, 35 are PNU and 40 are ODM

supporters. The percentage of NARC, PNU and ODM supporters who

read the Standard newspaper are 70%, 50% and 80% respectively.

Page 2 of 4

i)

If one person among the supporters is picked at random, what is

the

probability

the

he/she

reads the newspaper.

[2marks]

ii)

If one of the 100 people is observed reading The Standard, what is

the probability that he/she is an NARC supporter? [3marks]

b)

A and B are two events such that

1

P( )

A =

3

P A ? B =

3 and (

)

4 . Find

P(A | B where A | B = A n B if A and B are independent. [5marks]

c)

343 patients suffer from a certain disease. 96 of them were given a new

treatment and 14 of them cured. Of the untreated patients, 32 of them got

cured. If one patient is picked at random and is found to be cured, what is

the probability that he/se was not treated.

[5marks]

d)

A die is tossed four times. What is the probability that at least one’4’ will

occur.

[5marks]

Q4.

a)

The number of hours which ten students, taken in random, studied for an

examination are shown below.

No. of

8 6 11 13 10 5 18 15 2 9

hours

Grade in

56 44 79 72 70 54 94 85 33 65

examination

i)

Calculate the Spearman’s rank correlation coefficient.

[6marks]

ii)

What is the relationship between the number of hours studied and grade in

examination.

[2marks]

b)

The independent probabilities that the three sections X, Y and Z of a

costing department will encounter a computer error are: 0.1, 0.3, 0.3 each

week respectively. Determine the probability that:

i)

That there will be at least one computer error.

[3marks]

ii)

One and only one computer error will be encounter by the costing

department

next

week. [3marks]

Page 3 of 4

c)

The coefficient of rank correlation between marks in statistics and marks

in mathematics obtained by a certain group of students is 0.8, if the sum of

squares of the squares of the difference in ranks is given to be 13,

find the number of students in the group.

[6marks]

Q5.

Does the age at which a child begins to talk predict later score on a test of mental

ability? A study of the development of young children recorded the age in

months at which each of the 15 children spoke their first word and their Adaptive

Score, the result of an aptitude test taken much later. The data appear below.

Child 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Age 15 26 10 9 15 20 18 11 8 20 7 9 10 12 10

Score 95 71 82 91 102 87 93 100 104 94 113 96 83 84 102

i)

Plot a scatter diagram. What do you observe from the plot?

[4marks]

ii)

What is the correlation coefficient between the age and the Adaptive

scores?

[6marks]

iii)

Find the linear regression line that would predict the Adaptive score given

at any age.

[6marks]

iv)

Predict the score at age 19 and 21 months.

[4marks]

Page 4 of 4

UNIVERSITY EXAMINATIONS 2009/2010

INSTITUTIONAL BASED PROGRAMME (IBP)

EXAMINATION FOR THE DEGREE BACHELOR OF SCIENCE

SMA 160: PROBABILITY AND STATISTICS I

DATE:

Thursday 29th April 2010

TIME: 11.00a.m – 1.00p.m

INSTRUCTIONS: Answer Question one and any other two questions

Q1.

a)

Differentiate between the following terms

i)

Independent and dependent events

ii)

Mutually exclusive and mutually exhaustive events

iii)

Primary and Secondary sources of data

iv)

Continuous

and

discrete

variables

[8marks]

b)

A bag contains 5 red balls, 3 white balls and 7 blue balls. What is the

probability that a ball drawn at random will be,

i)

White

[1mark]

ii)

White or red

[2marks]

iii)

Neither

white

nor

blue [3marks]

c)

An incomplete distribution is given below

Variable 0-10 10-20 20-30 30-40 40-50

Frequency 14

f1 27 f2 15

If the median is 25 and the mode is 24 obtain the missing frequencies f1

and

f2.

Page 1 of 4

1

1

1

d)

Suppose

P( )

A = , P(B) =

and P(A n C) =

where A and C are

2

3

12

independent.

1

If P(B ? C) = show that B and C are mutually exclusive. [4marks]

2

e)

The first four moments of a distribution about the value 4 of a variable are

-1.5, 17,-30 and 108. Obtain the mean variance of the distribution.

[4marks]

f)

Given that the set (a, b, c ,d, e) has mean m and standard deviation s, write

down the mean and standard deviation of the set (a+k, b+k, d+k, e+k) in

terms of m and s respectively where k is a constant.

[2marks]

Q2.

Consider the following frequency distribution table.

Class

0-10

10-20 20-30 30-40 40-50 50-60

Interval

Frequency

15

17 19 27 19 12

a)

Find the mean and the standard deviation by changing both the location

(use an assumed mean of 35) and the scale (use common class width).

[7marks]

b)

Find

i)

Mode

ii)

Median

iii)

Interquartile

range

iv)

Coefficient

variation

v)

Coefficient

of

skewness

[13marks]

Q3.

a)

Out of 100 people, 25 are NARC, 35 are PNU and 40 are ODM

supporters. The percentage of NARC, PNU and ODM supporters who

read the Standard newspaper are 70%, 50% and 80% respectively.

Page 2 of 4

i)

If one person among the supporters is picked at random, what is

the

probability

the

he/she

reads the newspaper.

[2marks]

ii)

If one of the 100 people is observed reading The Standard, what is

the probability that he/she is an NARC supporter? [3marks]

b)

A and B are two events such that

1

P( )

A =

3

P A ? B =

3 and (

)

4 . Find

P(A | B where A | B = A n B if A and B are independent. [5marks]

c)

343 patients suffer from a certain disease. 96 of them were given a new

treatment and 14 of them cured. Of the untreated patients, 32 of them got

cured. If one patient is picked at random and is found to be cured, what is

the probability that he/se was not treated.

[5marks]

d)

A die is tossed four times. What is the probability that at least one’4’ will

occur.

[5marks]

Q4.

a)

The number of hours which ten students, taken in random, studied for an

examination are shown below.

No. of

8 6 11 13 10 5 18 15 2 9

hours

Grade in

56 44 79 72 70 54 94 85 33 65

examination

i)

Calculate the Spearman’s rank correlation coefficient.

[6marks]

ii)

What is the relationship between the number of hours studied and grade in

examination.

[2marks]

b)

The independent probabilities that the three sections X, Y and Z of a

costing department will encounter a computer error are: 0.1, 0.3, 0.3 each

week respectively. Determine the probability that:

i)

That there will be at least one computer error.

[3marks]

ii)

One and only one computer error will be encounter by the costing

department

next

week. [3marks]

Page 3 of 4

c)

The coefficient of rank correlation between marks in statistics and marks

in mathematics obtained by a certain group of students is 0.8, if the sum of

squares of the squares of the difference in ranks is given to be 13,

find the number of students in the group.

[6marks]

Q5.

Does the age at which a child begins to talk predict later score on a test of mental

ability? A study of the development of young children recorded the age in

months at which each of the 15 children spoke their first word and their Adaptive

Score, the result of an aptitude test taken much later. The data appear below.

Child 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Age 15 26 10 9 15 20 18 11 8 20 7 9 10 12 10

Score 95 71 82 91 102 87 93 100 104 94 113 96 83 84 102

i)

Plot a scatter diagram. What do you observe from the plot?

[4marks]

ii)

What is the correlation coefficient between the age and the Adaptive

scores?

[6marks]

iii)

Find the linear regression line that would predict the Adaptive score given

at any age.

[6marks]

iv)

Predict the score at age 19 and 21 months.

[4marks]

Page 4 of 4