May 2016

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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: PSYCHOLOGY

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

YEAR GROUP: ONE

DATE: WEDNESDAY, 2015 MAY 6

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. Inferential statistics is concerned with:

A. giving the best estimate of the population mean B. finding the measures of central tendencies

C. making conclusions about a population from sample data D. summarizing and presenting data

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

A. Distance B. Height C. Age D. Weight

3. Which of the following descriptive statistics is LEAST affected by extreme values?

A. Mean B. Median

C. Standard deviation D. Range

4. 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

5. Which of the following is a qualitative variable?

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

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The Council of Community Colleges of Jamaica Page 3 6. In a set of observations, the measure of central tendency that reports the value that occurs

MOST frequently is called the:

A. mean B. median C. mode D. range

Questions 7 – 10 refer to the information given in the table below:

7. The midpoint of the modal class is:

A. 59.5 B. 61 C. 63 D. 65

8. Determine the lower boundary of the third class interval

A. 55.5 B. 60.5 C. 59.5 D. 65.5

9. What is the class width?

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

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 4

56 - 60 6

61 - 65 7

66 -70 5

2

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The Council of Community Colleges of Jamaica Page 4 11. When constructing a histogram, the:

A. frequency and the upper limits are used B. frequency and the lower limits are used

C. frequency with the lower and upper boundaries are used D. cumulative frequency and the upper boundaries are used

Questions 12 – 15 refer to the following information:

The scores attained by a group of students on a Mathematics test are

66, 54, 62, 80, 59, 72, 60, 64, 81, 62

12. Determine the median score:

A. 62 B. 63 C. 64 D. 66

13. Determine the mean score:

A. 62 B. 64 C. 65 D. 66

14. Determine the range of the scores:

A. 27 B. 62 C. 63 D. 64

15. The interquartile range is:

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The Council of Community Colleges of Jamaica Page 5 16. If A and B are mutually exclusive events, then:

A. P A( B)0

B. P A( B)1

C. P A( )P B( )0

D. P A( B)0

17. If A and B are independent events, then:

A. P A( B)0

B. P A( B)P A( )P B( )

C. P A( )P B( )P A( B)

D. P A( B)0

18. If P A( B)0.6, P A( )0.45, P B( )0.35,then P A( B)

A. 0.1 B. 0.2 C. 0.6 D. 0.7

Question 19 and 20 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

19. The value of k is:

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

20. The expected value of X is:

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The Council of Community Colleges of Jamaica Page 6 21. Which of the following is NOT an assumption of 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

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 calls will be received within the hour?

A. 4 4e B. 4

8e C. 4

16e D. 2

16e

(Total 25 marks)

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THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER II – 2015 MAY

TIME: 1:00 P.M. – 4:00 P.M.

This Examination paper has 5 pages

INSTRUCTIONS:

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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

Calculate:

a. i. the mean (4 marks)

ii. the median (4 marks) iii. the mode (5 marks)

b. i. the variance (5 marks) ii. the 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

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The Council of Community Colleges of Jamaica Page 9 b. Two events A and B are such that P A( 1)0.25 and P B( )0.35. If A and B are independent events, find:

i. P A( ) (2 marks) ii. P A( B) (3 marks) iii. P A( B) (4 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

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

c. Using your graph, determine the:

i. median (2 marks)

ii. inter-quartile range (5 marks) iii. probability that a customer selected at random waited at least 40 minutes

(3 marks)

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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 intends to specialize in Accounts? (3 marks)

iii. Given that the student is from College C, what is the probability that they intend 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 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

a. 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)

iii. Calculate Var (X) (2 marks)

Specializations College A College B College C

Accounts 25 15 22

Finance 10 20 13

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The Council of Community Colleges of Jamaica Page 11 b. A continuous random variable follows a normal distribution with a mean of 120 km and a variance of 38 km.

i. Calculate P (x ≤ 140) (4 marks)

ii. Calculate P (110 ≤ x ≤ 140) (5 marks)

iii. At what value of x will there be 70% of the data? (5 marks)

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THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

ASSOCIATE OF SCIENCE EXAMINATION

SEMESTER II – 2015 MAY

TIME: 1:00 P.M. – 4:00 P.M.

SOLUTIONS SECTION A

1 C

11 C

21 C

31

41

2 C

12 B

22 C

32

42

3 B

13 D

23 B

33

43

4 D

14 A

24 B

34

44

5 A

15 A

25 B

35

45

6 C

16 D

26

36

46

7 C

17 B

27

37

47

8 B

18 B

28

38

48

9 B

19 A

29

39

49

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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

= - 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

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The Council of Community Colleges of Jamaica Page 14 = 1.8

iii. Var (X) = ∑x2_P(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)

i. P(A) =1- P(A1) =1- 0.25 = 0.75

ii. 𝑃(𝐴 ∩ 𝐵) = P(A)*P(B) = 0.75 *0.35 = 0.2625

iii. 𝑃(𝐴 ∪ 𝐵)_{= P(A) + P(B) - P(A∩ B)}

𝑃(𝐴 ∪ 𝐵)_{= 0.75 + 0.35 - 0.2625} = 1.1 – 0.2625

= 0.8375

[Question 3]

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

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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) = /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

X

P

6 64 4  e

= 0.195

ii.

P

(

X



1 )



1 

P

(

x



0 )

= 1 – 0.0183 = 0.9817

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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

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Question 5

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

i.

P

(

X



3 )



6

C

₃

p

3

q

3

= 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

₄

p

4

q

2 + 5 5 1 6

q

p

C

+ 6 6 0

6

q

p

C

= 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

b)

i. P (X ≤ 140) = ₎

16 . 6

120 140

(z 

P = ₎ 16 . 6 20 (z P

= P(z 3.25)

= Φ(3.25)

= 1

ii. P (110 ≤ x ≤ 140) = ₎

16 . 6 120 140 16 . 6 120 110

(  z  

P = ₎ 16 . 6 20 16 . 6 10 ( z P

= P(1.62z3.25)

= Φ(3.25) - Φ(-1.62) = Φ(3.25) - (1- Φ(1.62) = 1 – (1- 0.9474)

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The Council of Community Colleges of Jamaica Page 18 iii. Given that P (X< ? ) = ₎

16 . 6

120 ? (z 

P = 0.7

Let z =

16 . 6

120 ?

P ( Z < z) = 0.7

Φz = 0.7 z = Φ-1 0.7 z = 0.524

16 . 6

120

? _{= 0.524}