Dec 2015

Affix your signature below and return Section A to the invigilator before leaving the examination room. Failure to do so may result in disqualification.

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THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER I – 2015 DECEMBER

PROGRAMMES: CRIMINAL JUSTICE

COURSE NAME: STATISTICS FOR SOCIAL SCIENCES CODE : (MATH1208)

YEAR GROUP: ONE

DATE: MONDAY, 2015 DECEMBER 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.

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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. Which of the following is NOT a feature of inferential statistics?

A. Making decisions B. Drawing conclusions

C. Forecasting values of a population D. Summarizing data from a sample

2. Which of the following is an example of a discrete random variable?

A. Number of students on roll in a college B. Height of students in a club

C. The daily temperature for the past month D. Weight of the students in a class

3. All of the following are data collection methods EXCEPT:

A. interview B. observation C. sample D. survey

4. Which of the following is a qualitative variable?

A. The shoe size of students in a class

B. The marital status of employees in a company C. The scores of students on a Statistics test D. The height of athletes in a competition

5. A man spent 40% of his salary on food. How many degrees would that represent on a pie chart?

A. 400 B. 600 C. 1440

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6. Which of the following is the MOST APPROPRIATE graphical display to represent information in a frequency distribution comprising of eight (8) class intervals of equal size?

A. Bar chart B. Histogram C. Line graph D. Pie Chart

Use the information given in the table below to answer questions 7 – 9.

7. Determine the upper limit of the modal class.

A. 125 B. 129 C. 124.5 D. 129.5

8. Determine the lower boundary of the fourth class interval.

A. 135 B. 134.5 C. 137 D. 139.5

9. What is the class mark for the class 130-134?

A. 5 B. 130 C. 132 D. 134.5

10. What is the median for the set of values: 60, 45, 76, 80, 94, 87, 54, and 64?

A. 76 B. 70 C. 80 D. 87

Height (cm) Frequency

120 - 124 6

125 - 129 9

130 - 134 8

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11. The mean of 15, 28, x, 35, and 40 is 30. What is the value of x?

A. 30 B. 32 C. 118 D. 150

12. Which of the following descriptive statistics is least affected by extreme values?

A. Mean B. Median

C. Standard deviation D. Range

13. What is the standard deviation of the population data 5, 3, 4, 4, 4?

A. 0.4 B. 0.632 C. 0.4 D. 4

14. Which of the following could NOT be a probability value?

A. 8 3

B. 89% C. 0.98

D. 5 6

15. 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

16. Two (2) events A and B are mutually exclusive. If P (AB) = 0.75 and the P (A) is 0.35, what is P (B)?

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17. The conditional probability of event A given that B has already occurred is written as:

A. 𝑃(𝐴 ∪ 𝐵) B. 𝑃(𝐴/𝐵) C. 𝑃(𝐴 ∩ 𝐵) D. 𝑃(𝐵/𝐴)

18. In a group of 50 athletes, 20 have won at least one medal in a previous Olympic. If an athlete is selected at random, what is the probability that the athlete has NOT won an Olympic medal?

A. 0.04 B. 0.06 C. 0.40 D. 0.60

19. Determine the value of k in the probability distribution below.

A. 0.1 B. 0.2 C. 0.3 D. 1.0

20. Which of the following measures is NOT a measure of dispersion?

A. Range

B. Standard deviation C. Quartile deviation D. Mean

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

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

C. The probability of success is always equal to the probability of failure D. Each trial is classified as a success or a failure

X 0 1 2 3

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22. Which of the following about the curve of a standard normal variable, z, is FALSE?

A. The mean is 0 B. The variance is 1

C. The curve touches the horizontal axis at infinity D. The curve is bell shaped

23. 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

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. Which of the following could NOT represent a correlation coefficient?

A. - 0.45 B. - 0.98 C. 1.00 D. 1.24

(Total 25 marks)

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THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER I – 2015 DECEMBER

PROGRAMMES: CRIMINAL JUSTICE

COURSE NAME: STATISTICS FOR SOCIAL SCIENCES CODE : (MATH1208)

YEAR GROUP: ONE

DATE: MONDAY, 2015 DECEMBER 14

TIME: 1:00 P.M. – 4:00 P.M.

DURATION: 3 HOURS

EXAMINATION TYPE: FINAL

This Examination paper has 5 pages

INSTRUCTIONS:

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SECTION B

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

Question 1

The following frequency distribution represents the time waited in a line (in minutes) by 50 customers at a bank before they were attended to by a bank teller.

Time waited (mins) # of customers

1 - 5 4

6 -10 6

11-15 16

16-20 9

21-25 8

26-30 7

a. Calculate:

i. the mean (4 marks)

ii. the mode (4 marks)

iii. the median (5 marks)

b. Calculate:

i. the variance (5 marks)

ii. the standard deviation (2 marks)

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

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Question 2

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

𝑥 1 2 3 4 5

𝑃(𝑋 = 𝑥) 0.2 0.30 k 0.15 0.25

i. What is the value of the unknown probability k? (2 marks)

ii. What is the expectation of the distribution, E(X)? (4 marks)

iii. Calculate the variance, Var (X) (5 marks)

iv. Calculate the standard deviation (2 marks)

v. Determine 𝑃(1 ≤ 𝑥 < 3) (2 marks)

b. Events A and B are such that P (A) = 0.3 and P(B) = 0.45. If events A and B are independent events, find:

i. P (A∩B) (2 marks)

ii. P (A∩B1 _{) (3 marks)}

iii. P (A1_∩B1_{) (3 marks)}

iv. P(AB) (2 marks)

(Total 25 marks)

Question 3

a. Studies have shown that 30% of all patients taking a certain antibiotic will get a headache. Determine the probability that among 10 patients taking this antibiotic:

i. none will get a headache (3 marks)

ii. more than two (2) will get a headache (4 marks)

iii. at least four (4) will get a headache (5 marks)

iv. the expected number of patients that will get a headache (2 marks)

b. The average number of persons that uses a particular ATM machine is 6 per hour. What is the probability that:

i. eight (8) persons will use the machine in an hour (2 marks) ii. at least two (2) persons will use the machine in an hour (5 marks)

iii. three (3) persons will use the machine in half an hour (4 marks)

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Question 4

a. The following represents the gender and marital status of 80 employees in a particular company.

If an employee is chosen at random what is the probability that:

i. he/she is married (2 marks)

ii. it is a female (2 marks)

iii. it is a female who is single (2 marks)

iv. it is a male given that the employee is divorced (3 marks)

v. the personis married given that it is a female (3 marks)

b. It is assumed that a particular battery manufactured by a company follows a normal distribution with a mean lifetime of 800 hours and a standard deviation of 60 hours. Find the probability that a random sample of 64 batteries taken from a production batch will have a mean lifetime of:

i. less than 785 hours (4 marks)

ii. more than 820 hours (4 marks)

iii. between 800 and 810 hours (5 marks)

(Total 25 marks)

Single Married Divorced

Male 18 13 4

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Question 5

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

X 11 12 14 14 15 17 18 19

Y 11 13 14 16 18 19 21 24

a. Plot a scatter diagram to represent the data given. (5 marks)

b. Calculate the correlation coefficient and interpret the results. (8 marks)

c. Calculate the coefficient of determination of the results (2 marks)

d. Find the regression line equation in the form y = a +bx (5 marks)

e. Draw the line of best fit on the scatter diagram in (a) above (3 marks)

f. Use the regression line to predict the value of y when x = 25 (2 marks)

(Total 25 marks)

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THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER I – 2015 DECEMBER

PROGRAMMES: CRIMINAL JUSTICE

COURSE NAME: STATISTICS FOR SOCIAL SCIENCES CODE : MATH1208

YEAR GROUP:

DATE:

TIME:

DURATION: 3 HOURS

EXAMINATION TYPE: FINAL

SOLUTIONS

SECTION A

1

D

11

B

21

C

31

41

2

A

12

B

22

C

32

42

3

D

13

C

23

C

33

43

4

B

14

D

24

B

34

44

5

C

15

A

25

D

35

45

6

B

16

C

26

36

46

7

B

17

B

27

37

47

8

C

18

D

28

38

48

9

C

19

A

29

39

49

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SECTION B

Question 1

Mean = x = ∑fx = 810 = 16.2 ∑f 50

Mode = L1 + D1_ C

D1 + D2

= 10.5 +( 10/10+7 ) 5 =10.5 + (10/17) 5

= 10.5 + 2.94 = 13.44

Median= L1 + (N/2 -∑F1/fm) C

= 10.5 + (50/2 -10)/16) 5 = 10.5 + (15/16) 5 = 10.5 +4.69 = 15.19

Variance= ∑fx2_{/ ∑f – x}2 _{= 15 760/ 50 – (16.2)}2

= 315.2 – 262.44 = 52.76

Standard deviation= √∑fx2_{/∑f– x}2

=√52.76 = 7.26

Skewness = 3(mean – median) /standard deviation) = 3(16.2 – 15.19)/7.26

=3(1.01)/7.26 = 0.417

The distribution is positively skewed X

Midpoint

F

Frequency

Fx X2 Fx2

3 4 12 9 36

8 6 48 64 384

13 16 208 169 2704

18 9 162 324 2916

23 8 184 529 4232

28 7 196 784 5488

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[Question 2]

a) i. k = 1 – (0.2 + 0.3 + 0.15 + 0.25) = 1 – 0.9

= 0.1

ii. E(X) =∑xP(X=x) = (1*0.2)+(2*0.3) + (3*0.1) + (4*0.15) +(5*0.25) = 0.2 + 0.6 + 0.3 + 0.6 + 1.25

= 2.95

iii. Var (X) = ∑x2 – [E(X)]2

∑x2 = (1*0.2)+(4*0.3) + (9*0.1) + (16*0.15) +(25*0.25) = 0.4 + 1.2 + 0.9 + 2.4 + 6.25

= 10.95 Var (X) = 10.95 – (2.95)2 = 10.95 – 8.70 = 2.25

iv. Std. =√ 2.25 =1.5

v. 𝑃(1 ≤ 𝑥 < 3) = P(x=1) + P(x = 2) = 0.2 + 0.3

= 0.5

b) P(A) = 0.3 P(B) = 0.45

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[Question 3]

a) n = 10, p = 0.3 , q = 0.7

i.

P

(

X



0

)



C

p

q

= 1 x 0.30x 0.710

_{= 0.028}

ii. P (X > 2) = 1- [P (X = 0) + P (X = 1) + P (X = 2)]

7

.

0

3

.

0

)

1

(

X





C





P

= 10×0.3×0.04 = 0.12

7

.

0

3

.

0

)

2

(

X





C





P

= 45×0.09×0.058 = 0.2349

P (X > 2) = 1- [ 0.028 + 0.121 + 0.2349] = 1- (0.3839)

= 0.616

iii. P (X ≥ 4) = 1- [P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)]

7

.

0

3

.

0

)

3

(

X





C





P

= 120×0.027×0.082 = 0.2657

P (X ≥ 4) = 1- [ 0.028 + 0.12 + 0.2349 + 0.2657] = 1- (0.6486)

= 0.3514 iv. E(X) = 10 ×0.3 = 3

b) i.

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6

!

1

)

1

(





e

X

P

)]

1

(

)

0

(

[

1

)

2

(

X







P

x





P

x



P

= 1 – 0.01738 = 0.98262 iii. 3 3

3

!

3

)

3

(





e

X

P

i. P (married) = 33/80

ii. P (Female) = 45/80 = 9/16

iii. P (Female and Single ) = 22/80 =11/40

iv. P(male/divorced) = 4/7

v. P(married/female) = 20/45

b)

i. P (X < 785) = ₎ 60

800 785 (z  P

= ₎ 60

15 (z P

= P(z0.25)

= 1-Φ(0.25)

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= 0.4013

ii. P (X > 820) = ₎ 60

800 820 (z   P

= ₎ 60 20 (z P

= P(z 0.33)

= 1-Φ(0.33)

= 1- 0.6293

= 0.3707

ii. P (800< x < 810) = ₎

60 800 810 60

800 800

(  z 

P

= ₎

60 10 0

( z P

= P(0 z0.17)

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Question 5

a)

b)

X Y XY X2 Y2

11 11 121 121 121

12 13 156 144 169

14 14 196 196 196

14 16 224 196 256

15 18 270 225 324

17 19 323 289 361

18 21 378 324 441

19 24 456 361 576

∑X= 120 ∑Y=136 ∑XY= 2124 ∑X2_{= 1856 ∑Y}2_{= 2444}

The Council of Community Colleges of Jamaica Page 19

c)

r





0

.

977



d) b = 8(2124) – 120 x 136 8(1856) – (120)2

= 16992 - 16320 14848 -14400

= 672 448

= 1.5

















8

120

5

.

1

8

136

a

a = 17 – 1.5(15) a = -5.5

y = a + bx

y = -5.5+1.5x

e)

f) when x = 25