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# Introduction To Business Statistics Question Paper

Introduction To Business Statistics

Course:

Institution: Strathmore University question papers

Exam Year:2008

STRATHMORE UNIVERSITY

FACULTY OF COMMERCE

Bachelor of Commerce

END OF SEMESTER EXAMINATION

MAT 2102:

DATE: 8th October 2008 TIME: 2 Hours

INSTRUCTIONS: Answer Question ONE and any other TWO questions.

QUESTION ONE (30 marks)

(a) Define the following terms (i) Statistics (ii) Mutually exclusive events (iii) random

variable (iv) conditional probability

(6 marks)

(b) In a survey of a sample of interest rates on mortgages for 15-year mortgages at local

lending banks was found to be: 7.1%, 7.3%, 7.0%, 6.9%, 6.6%, 6.9%, 6.5%, 7.3%, 6.8%

Calculate and interpret (i) the mean (ii) median (iii) mode (iv)standard deviation

(4 marks)

(c) (i) Gogni Construction Company pays its hourly employees KSh. 65, 75 or 85 per

hour. There are 26 hourly employees; 14 are paid at 65 rate, 10 are paid at 75 and 2 are

paid at 85 rate.

Find the mean hourly rate paid to the employees. (2 marks)

(ii) Kamau’s Orchard sells apples in a large bag by weight. A sample of seven bags

contained the following numbers of apples: 23, 19, 26, 17, 21, 24 22. Compute the (i)

mean number of apples in a bag (ii) the median number of apples in a bag (iii) verify that

S(x - x) = 0 (4 marks)

(d) Let A and B be any events with P(A) = 1/2 and P(AUB) = 2/3,

Find (i) P(B) (ii) P(A/B) (iii) P(Bc) (iv) P(AcnBc) (6 marks)

(e) Given the following discrete probability distribution

x: 0 1 2 3

p(x): 0.2 0.4 0.3 0.1

Compute (i) the mean (ii) variance (iii) standard deviation (4 marks)

(f) Suppose the mean and standard of a distribution are given as 25 and 9 respectively,

Find the Z values of

(i) P(19 < X< 28)

(ii) P(X< 4) (4 marks)

QUESTION TWO (20 marks)

(a) ) Listed below is the number of 30-second radio advertisements sports purchased

by each of the 45 members of the Kenya Automobile Dealers Association last year:

96 93 88 117 127 95 113 96 139 142 94 197 125 155 155 103 112 135 132 111 125

104 106 139 118 136 125 143 120 103 113 124 108 94 146 156 112 127 117 120 134

119 97 89 138

(i) Organize the data into stem-and-leaf display

(ii) Around what values do the numbers of advertising sports tend to cluster?

(iii) What is the lowest and highest number of sports purchased by the dealers?

(6 marks)

(b) The following is a frequency distribution of the selling prices of vehicles sold at

Kisumu Motors Ltd last month:

Selling prices (K£’000) Frequency

12 – 15 10

16 – 18 20

19 – 21 16

22 – 24 19

25 – 27 8

28 – 30 4

31 – 33 2

34 – 36 1

(i) Construct a histogram and a frequency polygon

(ii) Construct a cumulative frequency curve (Ogive)

(iii) Estimate and interpret the median and the mode of the distribution from the

graphs

(12 marks)

QUESTION THREE (20 marks)

(a) The probability density function for a continuous random function X is given by

kx x

f x

0,

, 0 3

( )

Find (i) k

(ii) E(X)

(iii) Var (X) (8 marks)

(b) John Racho sells new cars for Baraka Motors ltd. John usually sells the large number

of cars on Saturday. He has established the following probability distribution for the

number of cars he expects to sell on a particular Saturday:

No. of cars sold Probability, p(x)

0 0.10

1 0.20

2 0.30

3 0.30

4 0.10

Required: (i) What Type of distribution is this?

(ii) On a typical Saturday, how many cars should John expect to sell?

(ii) What is the variance and standard deviation of the distribution?

(6 marks)

(c) The yearly income for a group of 10,000 professionals is normally distributed with

mean = K£ 60,000 and standard deviation = K£5,000.

(i) What is the probability that a person from the group has a yearly income of

less than K£ 56,000?

(ii) How many of the people have yearly income of over K£70,000?

(6 marks)

QUESTION FOUR (20 marks)

(a) A sample of amount of money paid for a film show on Saturday in a City Cinema

was organized into a frequency distribution as follows:

Amount paid Number of cars

30 – 75 8

75 – 100 10

100 – 125 32

125 – 150 40

150– 175 30

175– 200 18

200 – 225 8

225 – 250 4

Calculate

(i) quartile range

(ii) mean amount paid

(iii) mean absolute deviation

(iv) variance and (v) standard deviation. (12 marks)

(b) Suppose that the probability of a boy at each birth is 0.5. In families with three

children of which two are boys, what is the probability that the eldest child is a boy?

(8 marks)

QUESTION FIVE (20 marks)

(a) An insurance company manages is concerned about the health of female adults,

since the company is prepared to give a reduced premium rate to those who have a

certain level of fitness. In particular he would like to investigate how their heights are

related to their weights with a view to possibly using these measurements as a fitness

criterion. Given below is measurement of 12 female weights (kg) and heights (cm):

Female

No

1 2 3 4 5 6 7 8 9 10 11 12

Weight

(kg)

(y)

167 168 163 165 160 156 169 166 162 158 168 168

Height

(cm)

(x)

71.8 72.0 69.3 70 64.2 58.1 74.0 70.0 59.3 59.0 67.1 64.0

Required:

(i) Calculate and interpret the Pearson product moment correlation

coefficient, r

(ii) Find R2 and interpret

(iii) Find the trend-line

(iv) Find the weight when the height is 70 cm. (10 marks)

(b) (i) Name and define the components of a time series

(ii) An underpaid worker has supplemented his income over the years by

waiting on tables during the weekends in his local restaurant. The amount of

extra income that he has made by this activity is shown below:

Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Extra

Income

4701 5298 5938 6673 7209 7422 7780 8476 9066 9363 9885

Required:

(i) Find the secular trend of these date using the method of semiaverage,

and draw it onto a graph of data

(ii) Estimate extra income the worker will earn in 2007, 2008

(iii) Construct 3-year and 5-year moving averages and draw them on a

graph (10 marks)

FACULTY OF COMMERCE

Bachelor of Commerce

END OF SEMESTER EXAMINATION

MAT 2102:

DATE: 8th October 2008 TIME: 2 Hours

INSTRUCTIONS: Answer Question ONE and any other TWO questions.

QUESTION ONE (30 marks)

(a) Define the following terms (i) Statistics (ii) Mutually exclusive events (iii) random

variable (iv) conditional probability

(6 marks)

(b) In a survey of a sample of interest rates on mortgages for 15-year mortgages at local

lending banks was found to be: 7.1%, 7.3%, 7.0%, 6.9%, 6.6%, 6.9%, 6.5%, 7.3%, 6.8%

Calculate and interpret (i) the mean (ii) median (iii) mode (iv)standard deviation

(4 marks)

(c) (i) Gogni Construction Company pays its hourly employees KSh. 65, 75 or 85 per

hour. There are 26 hourly employees; 14 are paid at 65 rate, 10 are paid at 75 and 2 are

paid at 85 rate.

Find the mean hourly rate paid to the employees. (2 marks)

(ii) Kamau’s Orchard sells apples in a large bag by weight. A sample of seven bags

contained the following numbers of apples: 23, 19, 26, 17, 21, 24 22. Compute the (i)

mean number of apples in a bag (ii) the median number of apples in a bag (iii) verify that

S(x - x) = 0 (4 marks)

(d) Let A and B be any events with P(A) = 1/2 and P(AUB) = 2/3,

Find (i) P(B) (ii) P(A/B) (iii) P(Bc) (iv) P(AcnBc) (6 marks)

(e) Given the following discrete probability distribution

x: 0 1 2 3

p(x): 0.2 0.4 0.3 0.1

Compute (i) the mean (ii) variance (iii) standard deviation (4 marks)

(f) Suppose the mean and standard of a distribution are given as 25 and 9 respectively,

Find the Z values of

(i) P(19 < X< 28)

(ii) P(X< 4) (4 marks)

QUESTION TWO (20 marks)

(a) ) Listed below is the number of 30-second radio advertisements sports purchased

by each of the 45 members of the Kenya Automobile Dealers Association last year:

96 93 88 117 127 95 113 96 139 142 94 197 125 155 155 103 112 135 132 111 125

104 106 139 118 136 125 143 120 103 113 124 108 94 146 156 112 127 117 120 134

119 97 89 138

(i) Organize the data into stem-and-leaf display

(ii) Around what values do the numbers of advertising sports tend to cluster?

(iii) What is the lowest and highest number of sports purchased by the dealers?

(6 marks)

(b) The following is a frequency distribution of the selling prices of vehicles sold at

Kisumu Motors Ltd last month:

Selling prices (K£’000) Frequency

12 – 15 10

16 – 18 20

19 – 21 16

22 – 24 19

25 – 27 8

28 – 30 4

31 – 33 2

34 – 36 1

(i) Construct a histogram and a frequency polygon

(ii) Construct a cumulative frequency curve (Ogive)

(iii) Estimate and interpret the median and the mode of the distribution from the

graphs

(12 marks)

QUESTION THREE (20 marks)

(a) The probability density function for a continuous random function X is given by

kx x

f x

0,

, 0 3

( )

Find (i) k

(ii) E(X)

(iii) Var (X) (8 marks)

(b) John Racho sells new cars for Baraka Motors ltd. John usually sells the large number

of cars on Saturday. He has established the following probability distribution for the

number of cars he expects to sell on a particular Saturday:

No. of cars sold Probability, p(x)

0 0.10

1 0.20

2 0.30

3 0.30

4 0.10

Required: (i) What Type of distribution is this?

(ii) On a typical Saturday, how many cars should John expect to sell?

(ii) What is the variance and standard deviation of the distribution?

(6 marks)

(c) The yearly income for a group of 10,000 professionals is normally distributed with

mean = K£ 60,000 and standard deviation = K£5,000.

(i) What is the probability that a person from the group has a yearly income of

less than K£ 56,000?

(ii) How many of the people have yearly income of over K£70,000?

(6 marks)

QUESTION FOUR (20 marks)

(a) A sample of amount of money paid for a film show on Saturday in a City Cinema

was organized into a frequency distribution as follows:

Amount paid Number of cars

30 – 75 8

75 – 100 10

100 – 125 32

125 – 150 40

150– 175 30

175– 200 18

200 – 225 8

225 – 250 4

Calculate

(i) quartile range

(ii) mean amount paid

(iii) mean absolute deviation

(iv) variance and (v) standard deviation. (12 marks)

(b) Suppose that the probability of a boy at each birth is 0.5. In families with three

children of which two are boys, what is the probability that the eldest child is a boy?

(8 marks)

QUESTION FIVE (20 marks)

(a) An insurance company manages is concerned about the health of female adults,

since the company is prepared to give a reduced premium rate to those who have a

certain level of fitness. In particular he would like to investigate how their heights are

related to their weights with a view to possibly using these measurements as a fitness

criterion. Given below is measurement of 12 female weights (kg) and heights (cm):

Female

No

1 2 3 4 5 6 7 8 9 10 11 12

Weight

(kg)

(y)

167 168 163 165 160 156 169 166 162 158 168 168

Height

(cm)

(x)

71.8 72.0 69.3 70 64.2 58.1 74.0 70.0 59.3 59.0 67.1 64.0

Required:

(i) Calculate and interpret the Pearson product moment correlation

coefficient, r

(ii) Find R2 and interpret

(iii) Find the trend-line

(iv) Find the weight when the height is 70 cm. (10 marks)

(b) (i) Name and define the components of a time series

(ii) An underpaid worker has supplemented his income over the years by

waiting on tables during the weekends in his local restaurant. The amount of

extra income that he has made by this activity is shown below:

Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Extra

Income

4701 5298 5938 6673 7209 7422 7780 8476 9066 9363 9885

Required:

(i) Find the secular trend of these date using the method of semiaverage,

and draw it onto a graph of data

(ii) Estimate extra income the worker will earn in 2007, 2008

(iii) Construct 3-year and 5-year moving averages and draw them on a

graph (10 marks)

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