May 2014

The Council of Community Colleges of Jamaica Page 1

THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER II – 2015 MAY

PROGRAMMES: SOCIAL WORK

COURSE NAME: STATISTICS FOR SOCIAL WORKERS CODE : (MATH1203)

YEAR GROUP: ONE

DATE: THURSDAY, 2015 MAY 14

TIME: 1:00 P.M. – 4:00 P.M.

DURATION: 3 HOURS

EXAMINATION TYPE: FINAL

This Examination paper has 6 pages INSTRUCTIONS:

SECTION A: ANSWER ALL QUESTIONS IN THIS SECTION.

SECTION B: ANSWER ANY THREE (3) QUESTIONS FROM THIS SECTION.

The Council of Community Colleges of Jamaica Page 2

SECTION A

Instructions: On the computerized answer sheet provided, shade the letter that corresponds with the most appropriate response for each of the following.

1. The type of statistics that deals with organizing and summarizing data is:

A. classical statistics B. descriptive statistics C. general statistics D. inferential statistics

2. Which of the following is NOT an example of a continuous variable?

A. Height of students in your class

B. Ages of the students enrolled in college C. The daily temperature for the past month D. Weight of the lecturers at your college

3. All of the following are advantages of using questionnaires to collect data EXCEPT:

A. quick turnaround time

B. can be administered to groups

C. ease of data analysis for closed-ended items D. allows probing and follow up- questions

4. Which of the following is a qualitative variable?

A. The weights of students in your class B. The colours of vehicles in the car park

C. The scores of your students on a Statistics test D. The height of the first year students at your college

5. The term used to describe data values that are not symmetric and tend to be more on one side than the other is:

A. binomial B. normal C. poisson D. skewness

The Council of Community Colleges of Jamaica Page 3 Questions 6 – 10 refer to the information given in the table below:

6. Determine the modal class:

A. 51 – 55 B. 56 – 60 C. 60 – 65 D. 61 – 65

7. Determine the upper boundary of the third class interval

A. 54.5 B. 55.5 C. 59.5 D. 65.5

8. What is the class width?

A. 4 B. 5 C. 6 D. 10

9. Calculate the mean weight.

A. 27 B. 58 C. 61 D. 63

10. Which of the following would be the BEST diagram to represent the data in the table?

A. Bar chart B. Histogram C. Pictogram D. Pie Chart

Weight (kg) Frequency

51 - 55 5

56 - 60 7

61 - 65 9

The Council of Community Colleges of Jamaica Page 4 Questions 11 – 13 refer to the following information below:

The following represents the scores of 10 students in a shooting competition

5, 6, 9, 4, 5, 8, 3, 6, 9, 5

11. Determine the median score

A. 5.5 B. 6 C. 10 D. 60

12. Determine the mean score

A. 3 B. 5 C. 6 D. 9

13. Determine the range of the scores

A. 3 B. 5 C. 6 D. 9

14. If 𝑃(𝐴 ∩ 𝐵) = 0, then

A. A and B are mutually exclusive B. A and B are independent

C. A and B are complementary events D. A and B are equal events

15. Two (2) events C and D are such that P (C / D) = P (C). What can be said about the two events?

The Council of Community Colleges of Jamaica Page 5 16. If P (C) = 0.05, what is P (C1) ?

A. 0.05 B. 0.50 C. 0.95 D. 1.05

Question 17 and 18 relate to the following information:

A random variable X has the probability distribution shown in the table below:

X 0 1 2 3

P(X) 0.1 k 0.4 0.3

17. The value of k is:

A. 0.2 B. 0.3 C. 0.4 D. 1

18. The expected value of X is:

A. 1.7 B. 1.9 C. 2.26 D. 4.6

19. Which of the following is NOT a measure of dispersion?

A. Range

B. Standard deviation C. Quartile deviation D. Mean

20. Which of the following is NOT true about the Binomial distribution?

A. Each trial is identical B. Each trial is independent

The Council of Community Colleges of Jamaica Page 6 21. In its standardized form, the normal distribution:

A. cannot be used to approximate discrete probability B. has a mean of 1 and a variance of 0

C. has a mean of 0 and a standard deviation of 1 D. has an area equal to 0.5

22. The random variable X is normally distributed with mean 12 and standard deviation 0.5. What is the probability that X < 12?

A. 0 B. 1 C. 0.5 D. -1

23. The standard deviation of a set of data is zero (0), this means:

A. the mean is 0

B. all the values are the same C. the variance is 1

D. the range is greater than 1

24. Find the variance of a binomial distribution when n = 12 and p = 0.4

A. 1.70 B. 2.88 C. 4.80 D. 9.12

25. The number of calls received by a telephone operator during an hour has a Poisson distribution with an average of 4 calls. What is the probability that exactly two (2) calls will be received within the hour?

A. 4e-4 B. 8e-4 C. 16e-4 D. 16e-2

(Total 25 marks)

The Council of Community Colleges of Jamaica Page 7

THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER II – 2015 MAY

PROGRAMMES: SOCIAL WORK

COURSE NAME: STATISTICS FOR SOCIAL WORKERS CODE : (SOWK1203)

YEAR GROUP: ONE

DATE: THURSDAY, 2015 MAY 14

TIME: 1:00 P.M. – 4:00 P.M.

DURATION: 3 HOURS

EXAMINATION TYPE: FINAL

This Examination paper has 5 pages INSTRUCTIONS:

The Council of Community Colleges of Jamaica Page 8

SECTION B

Instructions: Answer any THREE (3) questions from this section.

Question1

The following table gives the frequency distribution of the number of calls received by a company at different intervals within the first hour of opening on a particular day.

Intervals (mins) Number of calls

1 - 10 4

11-20 6

21-30 14

31-40 16

41-50 12

51-60 8

a. Calculate the:

i. mean (4 marks)

ii. mode (4 marks)

iii. median (5 marks)

b. Calculate the:

i. variance (5 marks)

ii. standard deviation (2 marks)

c. Calculate the skewness and interpret the results (5 marks)

(Total 25 marks)

Question 2

a. Let 𝑋 be the random variable with the probability distribution given by:

𝑥 0 1 2 3 4

𝑃(𝑋 = 𝑥) 0.3 𝑏 0.15 0.1 0.25

The Council of Community Colleges of Jamaica Page 9 b. Let X be a discrete random variable that follows a binomial distribution with n = 6 and p = 0.7.

i. Calculate the probability of obtaining exactly 3 successes (4 marks)

ii. Calculate the probability of obtaining at least 4 successes (5 marks)

(Total 25 marks)

Question 3

The following shows the results of a survey of the number of minutes customers waited in line before being attended to by a cashier at a commercial bank.

Wait time (mins) Number of customers

1 - 10 2

11-20 6

21-30 12

31-40 16

41-50 8

51-60 4

61-70 2

a. Construct a less than cumulative frequency table (7 marks)

b. Draw a less than cumulative frequency curve to represent the data above (Use a scale of

1 cm to represent 5 units on both axes) (8 marks)

c. Using your graph, determine the:

i. median (3 marks)

ii. inter-quartile range (4 marks) iii. probability that a customer selected at random waited at least 40 minutes

(3 marks)

The Council of Community Colleges of Jamaica Page 10

Question 4

a. A survey was done among 175 first year students in three different community colleges to determine their intended area of specialization. The results are listed in the table below.

i. If a student is chosen at random, what is the probability that they are from college A? (3 marks)

ii. What is the probability that a randomly selected student intend to specialize in Accounts? (3 marks)

iii. Given that the student is from College C, what is the probability that he/she intends to specialize in Management. (3 marks)

b. The number of cars passing through a toll booth per hour follows a Poisson distribution with a mean of 4. Calculate the probability that:

i. exactly 3 cars passed during a particular hour (3 marks)

ii. one or more cars will pass during the an hour (3 marks)

iii. at least two cars will pass during the hour (6 marks)

c. Identify two (2) strengths and two (2) weaknesses of using questionnaires as a data collection instrument. (4 marks)

(Total 25 marks)

Question 5

The table below shows data collected from two variables X and Y

X 7 9 10 15 11 17 19 22

Y 8 12 14 13 15 16 17 18

a. Calculate and interpret r and r2 (12 marks)

Specializations College A College B College C

Accounts 25 15 22

Finance 10 20 13

The Council of Community Colleges of Jamaica Page 11 b. Calculate the value of the slope of the regression line. (4 marks)

c. Calculate the value of the y intercept of the regression line. (5 marks)

d. Using the regression equation, predict the value of Y when X =30 (4 marks)

(Total 25 marks)

The Council of Community Colleges of Jamaica Page 12

THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER II – 2015 MAY

PROGRAMMES: SOCIAL WORK

COURSE NAME: STATISTICS FOR SOCIAL WORKERS CODE : (SOWK1203)

YEAR GROUP: ONE

DATE: THURSDAY, 2015 MAY 14

TIME: 1:00 P.M. – 4:00 P.M.

DURATION: 3 HOURS

EXAMINATION TYPE: FINAL

SOLUTIONS SECTION A

1

B

11

A

21

C

31

41

2

B

12

C

22

C

32

42

3

D

13

C

23

B

33

43

4

B

14

A

24

B

34

44

5

D

15

B

25

B

35

45

6

C

16

C

26

36

46

7

D

17

A

27

37

47

8

B

18

B

28

38

48

9

C

19

D

29

39

49

The Council of Community Colleges of Jamaica Page 13

SECTION B

Question 1

Mean = x = ∑fx = 2030 = 33.8 ∑f 60

Mode = L1 + D1_ C

D1 + D2

= 30.5 +( 2/2+4 ) 10 =30.5 + (2/6) 10

= 30.5 + 3 = 33.5

Median= L1 + (N/2 -∑F1/fm) C

= 30.5 + (60/2 -20)/16) 10 = 30.5 + (10/16) 10 = 30.5 + 6.25 = 36.75

Variance= ∑fx2_{/ ∑f – x}2 _{= 80315/ 60 – (33.8)}2

= 1338.58 – 1142.44 = 196.14

Standard deviation= √∑fx2_{/∑f– x}2

=√196.14 = 14

Skewness = 3(mean – median) /standard deviation) = 3(33.8 - 36.75)/14

=3(-2.95)/14 = - 8.85/14 =

X

Midpoint

F

Frequency

Fx X2 Fx2

5.5 4 22 30.25 121

15.5 6 93 240.25 1441.5

25.5 14 357 650.25 9103.5

35.5 16 568 1260.25 20164

45.5 12 546 2070.25 24843

55.5 08 444 3080.25 24642

The Council of Community Colleges of Jamaica Page 14 = - 0.63

The distribution is negatively skewed

[Question 2]

a) i. b = 1 – (0.3 + 0.15 + 0.1 + 0.25) = 1 – 0.8

= 0.2

ii. E(X) =∑xP(X=x) = (0*0.3)+(1*0.2) + (2*0.15) + (3*0.1) +(4*0.25) = 0 +0.2 + 0.3 + 0.3 + 1

= 1.8

iii. Var (X) = ∑x2P(X=x)

= (0*0.3)+(1*0.2) + (4*0.15) + (9*0.1) + (16*0.25) = 0 + 0.2 + 0.6 + 0.9 + 4

= 2.1 iv. Std. =√ 2.1 =1.45

v. 𝑃(1 ≤ 𝑥 < 3) = P(x=1) + P(x = 2) = 0.2 + 0.15

= 0.35

b) n = 6, p = 0.7 , q = 0.3

i.

P

(

X



3

)



C

p

q

= 20 x 0.73_{x 0.3}3 _{= 0.1852}

ii. P (X ≥ 4) = P (X = 4) + P (X = 5) + P (X = 6)

=6

C

p

q

C

p

q

C

p

q

= 15 x 0.74 x 0.32 +6 x 0.75 x 0.31 + 1 x 0.76x 0.30 = 0.3241 + 0.3025 +0.1177

= 0.7743 iii. Var (X) = npq

= 6 × 0.7 × 0.3 = 1.26

The Council of Community Colleges of Jamaica Page 15 a)

Wait time (mins) Number of customers

Class boundaries Cumulative frequency

1 - 10 2 0.5 - 10.5 2

11 - 20 6 10.5 - 20.5 8

21 - 30 12 20.5 - 30.5 20

31 - 40 16 30.5 - 40.5 36

41 - 50 8 40.5 - 50.5 44

51 - 60 4 50.5 - 60.5 48

61 - 70 2 60.5 - 70.5 50

b)

c) From Graph

i. median = ½ (50) = 25th reading ii. Q1 = ¼ (50) = 12.5th reading Q3 = ¾ (50) = 37.5th reading

ITQR = Q3-Q1

iii. P (X ≥ 40) = 13/50 = /50

Question 4

a)

i. P (college A) = 70/175

ii. P (Accounts) = 62/175

iii. P (Management/College C) = 5/40 =1/8

b)

i.

3 4

4

!

3

)

3

(







e

The Council of Community Colleges of Jamaica Page 16

6 64 4

 e

= 0.195

ii.

P

(

X



1

)



1



P

(

x



0

)

iii.

P

(

X



2

)



1





P

(

x



0

)



P

(

x



1

)



1 4

4

!

1

)

1

(







e

X

P

= 4e-4 = 0.0733

Therefore P(X ≥2) = 1 – (0.0183 + 0.0733) = 1 – ( 0.0916)

= 0.9084

c. Strengths of questionnaires  quick turnaround time

 can be administered to groups  good for measuring attitudes

 facilitates anonymity of participants

Weaknesses of questionnaires  usually must be kept short  nonresponse to selected items  does not allow follow up questions

Question 5

The Council of Community Colleges of Jamaica Page 17

X Y XY X2 Y2

7 8 56 49 64

9 12 108 81 144

10 14 140 100 196

15 13 195 225 169

11 15 165 121 225

17 16 272 289 256

19 17 323 361 289

22 18 396 484 324

∑X= 110 ∑Y=113 ∑XY= 1655 ∑X2_{= 1710 ∑Y}2_{= 1667}

_{2 2}

113

1667

9

110

1710

8

113

110

1655

8

















r

27

.

47

75

.

39

810





r

98

.

1878

810



r





2

43

.

0



r

The Council of Community Colleges of Jamaica Page 18 = 13240 - 12430

13680 -12100

= 810 1580

= 0.513

c)

















8

110

513

.

0

8

113

a

a = 14.125 – 0.513(13.75) a = 7.07

d) y = a + bx

y = 7.07+0.513x

therefore when x = 30

The Council of Community Colleges of Jamaica Page 19