Econ 245

Summer A01 2017

Page 1

UNIVERSITY OF VICTORIA

Midterm May 31, 2017

Solutions

NAME: _____________________________

STUDENT NUMBER: V00______________

Course Name & No. Descriptive Statistics and Probability

Economics 245 Section(s) A01

CRN: 31111

Instructor: Betty Johnson

Duration: 1hour 50 minutes

This exam has a total of _15_ pages including this cover page.

Students must count the number of pages and report any discrepancy immediately to the Invigilator.

This exam is to be answered: In Booklets provided

Marking Scheme: 1. 20 marks

2. 15 marks

3. 5 marks

4. 5 marks

5. 5 marks

6. 5 marks

7. 12 marks

8. 10 marks

9. 10 marks

10. 9 marks

11. 4 marks

Materials allowed: Non-programmable calculator

Econ 245

Summer A01 2017

Page 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the

question. Question 1: (20 marks)

1. Which of the following statements involve descriptive statistics as opposed to inferential statistics?

A) The Alcohol, Tobacco and Firearms Department reported that Houston had 1,791 registered gun

dealers in 1997.

B) Based on a survey of 400 magazine readers, the magazine reports that 45% of its readers prefer

double column articles.

C) The FAA samples 500 traffic controllers in order to estimate the percent retiring due to job stress

related illness.

D) Based on a sample of 300 professional tennis players, a tennis magazine reported that 25% of the

parents of all professional tennis players did not play tennis.

ANSWER: A

2. The width of each bar in a histogram corresponds to the

A) midpoint of the class

B) number of observations in the class

C) boundaries of the class

D) percentage of observations in the class

ANSWER: C

3. Which of the following is not a goal of descriptive statistics?

A) Summarizing data

B) Displaying aspects of the collected data

C) Reporting numerical findings

D) Estimating characteristics of the population

ANSWER: D

4. Which of the following are not typical uses of inferential statistics?

A) Develop forecasts

B) Summarize data

C) Provide estimates

D) Make predictions

ANSWER: B

5. A line graph that connects points that represent the cumulative percentage of observations below the

upper limit of each class interval in a cumulative frequency distribution is known as a(n):

A) ogive

B) frequency histogram

C) frequency polygon

D) scatter plot

ANSWER: A

Econ 245

Summer A01 2017

Page 3

6. The total area of bars in a relative frequency histogram:

A) at least 1

B) at most 1

C) exactly 1

D) any value between 0 and 1

ANSWER: C

7. In a frequency distribution, what is the number of observations in a class called?

A) Class midpoint

B) Class frequency

C) Class interval

D) None of the above

ANSWER: B

8. Suppose you are told that the mean sample of numbers is below the median. What does this information

suggest?

A) The distribution is symmetric.

B) The distribution is skewed to the right or positively skewed.

C) The distribution is skewed to the left or negatively skewed. D) There is insufficient information to determine the shape of the distribution.

ANSWER: C

9. If you are interested in comparing variation in sales for small and large stores selling similar goods,

which of the following is the most appropriate measure of dispersion?

A) The range

B) The interquartile range

C) The standard deviation

D) The coefficient of variation

ANSWER: D

10. A bored carpenter counts the actual number of nails in 10 boxes of nails and records his findings as:

254, 275, 302, 287, 312, 323, 301, 319, 342, and 298. What can we say about the shape of the

distribution of the number of nails?

A) Symmetric

B) Skewed to the right.

C) Approximately mound-shaped. D) Skewed to the left.

ANSWER: D

Econ 245

Summer A01 2017

Page 4

11. A professor collected data on the number of absences in an introductory statistics class of 100 students

over the course of a semester. The data are summarized below.

# of

Absences

0 1 2 3 4 5 6

# of Students 5 13 24 23 17 11 7

What is the weighted mean number of absences per semester?

A) 3.14

B) 2.0

C) 2.95 D) 3.07

ANSWER: C

12. A sample of 15 observations has a standard deviation of 4. The sum of the squared deviations from the

sample mean is:

A) 19

B) 56

C) 60

D) 224

ANSWER: D

THE NEXT FOUR QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:

The following data represent a sample of 10 scores on a statistics quiz: 16, 16, 16, 16, 16, 18, 18, 20, 20, and 20.

13. The mean score is

A) 17.6

B) 15.8

C) 12.2

D) 10.4

ANSWER: A

14. The median score is

A) 16

B) 17

C) 18

D) 19

ANSWER: B

15. The modal score is

A) 16

B) 17

C) 18

D) 20

ANSWER: A

Econ 245

Summer A01 2017

Page 5

16. The standard deviation of the scores is

A) 1.682

B) 2.174

C) 3.3782

D) 1.838

ANSWER: D

17. When extreme values are present in a set of data, which pair of the following descriptive summary

measures of central tendency and dispersion is most appropriate?

A) Mean and standard deviation

B) Median and interquartile range

C) Range and coefficient of variation

D) Mode and variance

ANSWER: B

18. The manager of a local RV sales lot has collected data on the number of RVs sold per month for the last

five years. That data is summarized below

# of Sales 0 1 2 3 4 5 6

# of

Months

2 6 9 13 21 7 2

What is the weighted mean number of sales per month?

A) 3.31

B) 3.23

C) 3.54 D) 3.62

ANSWER: B

The accompanying table shows a price index over 10 time periods. Suppose the government agency updates this

series by making period 6 the base period with a value of 100.

Period Index

1 100.0

2 113.0

3 119.0

4 124.0

5 125.0

6 131.0

7 133.0

8 144.0

9 147.0

10 151.0

Econ 245

Summer A01 2017

Page 6

19. What would be the revised price index for period 2?

A) 82.66

B) 86.26

C) 78.44

D) 76.56

ANSWER: B

20. What would be the revised price index for period 9?

A) 116.08

B) 122.48

C) 112.21

D) 118.62

ANSWER: C

Question 2 (15 Marks) TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.

1. Inferential statistics are used to make a forecast regarding future trends.

ANSWER: T

2. Descriptive statistics is the study of a sample that enables a researcher to make projections or

estimates about the population from which the sample is drawn.

ANSWER: F

3. A parameter is a descriptive measure computed from an entire population.

ANSWER: T

4. A line chart is also called a scatter plot.

ANSWER: F

5. Although the range measures the total spread of the data, the interquartile range (IQR) measures

only the spread of the middle 50% of the data.

ANSWER: T

6. The sample covariance must take a value between –1 and +1 inclusive.

ANSWER: F

7. The sample covariance may never be negative.

ANSWER: F

8. When the data values are arranged in an ascending order, the third quartile ( 3Q ) is located in the 75th

percentile position, and first quartile ( 1Q ) is located in the 25th

percentile position.

ANSWER: T

9. Since the interquartile range takes into account only two of the data values, it is susceptible to

considerable distortion if there is an unusual number of extreme observations (outliers).

ANSWER: F

Econ 245

Summer A01 2017

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10. Although range and interquartile range measure the spread of data, both measures take into account

only two of the data values, regardless of the size of the data.

ANSWER: T

11. For any symmetrical distribution, the standard deviation is equal to the variance.

ANSWER: F

12. For a set of numerical data values arranged in ascending order, the value of the observation in the

center is called the geometric mean.

ANSWER: F

13. The geometric mean of a set of 10 positive numbers is the 10th

root of the product of the 10 values.

ANSWER: T

14. A negatively skewed distribution is not symmetrical. The long tail is to the right.

ANSWER: F

15. If the population variance 2 is unknown, a sample variance 2s is a better estimator of 2 if the

denominator is 2s formula is (n-1) rather than n.

ANSWER: T

Question 3: SHORT ANSWER. (5 Marks)

What is the difference between a frequency distribution and a relative frequency distribution?

ANSWER:

The basic difference between the frequency and relative frequency distribution is that the relative

frequency distribution tells us the percentages rather than the actual values. This way the data is

standardized making it much easier to compare it to other samples and populations.

Question 4: SHORT ANSWER. (5 Marks)Why do we have so many different measures of central

tendency? Are all really necessary or do they essentially provide the same information?

ANSWER:

For certain types of well-behaved data, all the measures will provide very similar information. However,

because much data is not well-behaved, it is often beneficial to look at a number of measures of central

tendency and report the one that best describes the location and average of the data. Wherever possible,

the mean is the preferred measure of location, because it uses all the data values. However, for certain

types of data that contain severe outliers (like income data), the median is the preferable measure of

central tendency because the outliers do not distort it.

Econ 245

Summer A01 2017

Page 8

Question 5: SHORT ANSWER. (5 Marks) Why is it necessary for a measure of variation to

accompany a measure of central tendency?

ANSWER:

A measure of central tendency alone does not give a complete picture of the data set. The object of

summary measures is to visualize the data set based on these measures. Hence, the measure of central

tendency locates the data set, but a measure of variation completes the picture by describing the

dispersion in the data about the location measure.

Question 6: (5 Marks)

Prove that the population variance can be equal to the following:

2 2

1

2

1

21 1

N

XN

Xii

N

ii

N

( )

where

N = population size

2

1

2

2 2 2

1

2 2 2

1 11

2 2 2

1

2 2 2

1

12

12

12

12

NX Expand

NX X

NX X

NX N N

NX N

ii

N

i ii

N

ii

N

ii

N

i

N

ii

N

i

Take summation operator through

Since X

N

then: N = X

i

i

( )

N

NX N

NX

i

N

ii

N

ii

N

2

1

2 2 2

1

2 2

1

2

1

1

Econ 245

Summer A01 2017

Page 9 Question 7: (12 Marks)

The following table of data relates to the number of ants collected in a month by students in

several grade 5 classes:

Interval (# of ants) Frequency

0 < x 20 112

20 < x 40 114

40 < x 60 124

60 < x 80 148

80 < x 100 102

Compute the

(i) mode, mode=70, or 60 to 80

(ii) arithmetic mean, [1120+3420+6200+10360+9180]/600=50.4667

(iii) the median observation 300 and 301: 50, or 40 to 60

(iv) the population variance:

=[183405.7244+47752.8267+27.0044+56469.5644+159414.2133]/600

=447069.3333/600 = 745.1156

Sheppard’s correction: 745.1156-33.3333=711.7822

(v) the standard deviation 26.6792

(vi) coefficient of variation. %8651.521004667.50

6792.26100

cv

Question 8: (10 Marks)

(i) Determine a frequency distribution and histogram for the following data.

Let the class width be 5 units wide (4.99999…. actually) and start at 15. (5 marks)

(ii) Draw the Ogive for this data. (5 marks)

25 24 16 49 37 42

17 18 37 32 45 43

23 19 47 35 22 42

Sort data first

1) 16

2) 17

3) 18

4) 19

5) 22

6) 23

7) 24

Econ 245

Summer A01 2017

Page 10

8) 25

9) 32

10) 35

11) 37

12) 37

13) 42

14) 42

15) 43

16) 45

17) 47

18) 49

Class (i) Range ($)

(width =10)

Frequency f i

1 15 X<20 4

2 20 X<25 3

3 25 X<30 1

4 30 X<35 1

5 35 X<40 3

6 40≤X<45 3

7 45≤X<50 3

Frequency

4

3

2

1

(15 to 20) (20 to 25) (25 to 30) (30 35) (35≤40) (40 45) (45≤50)

Econ 245

Summer A01 2017

Page 11

Cumulative Frequency

18

Ogive

16

12

8

4

Question 9 (10 marks)

Campus Security has reviewed the number of parking citations issued per day by each of the 10

officers. The data were: 11, 19, 10, 32, 29, 11, 20, 24, 23, and 21.

a) What is the mean number of tickets issued per day?

µ=(200)/10=20

b) What is the median number of citations issued per day?

10

11

11

19

20

21

23

24

29

32

(20+21)/2=20.5

(15 to 20) (20 to 25) (25 - 30) (30- 35) (35 40) (40≤45) (45≤50)

Econ 245

Summer A01 2017

Page 12

c) What is the mode of the number of tickets issued per day?

11

d) What is the first quartile of the number of tickets issued per day?

P25=(N*K)/100

=(10*25)/100=2.5 location: round up to 3rd

observation: 11.

e) What is the interquartile of the number of tickets issued per day?

P75=(10*75)/100=7.5 location: round up to 8th

observation: 24

Interquartile range= 24-11=13

Question 10: (9 marks) (a) Use the following data to construct the Paasche price index for 2016, with a base value of

100 in 1995: (Table your answer with the two entries.) (3 Marks)

Good 1995

Price 1995

Quantity 2016

Price 2016

Quantity Toothpaste 1 24 3 25

Dental floss 2 5 4 5

4286.27110035

95

1001025

2075

10052251

54253

2016,1995

2016,1995

2016,1995

P

P

P

P

P

P

Year Index

1995 100

2000 271.4286

(b) Generate the Paasche quantity index with a base value of 100 in 1995. (3 marks)

2609.10392

95100

)4)(5()3)(24(

)4)(5()3)(25(0

P

tQ

(c) Determine if the Paasche index passes or fails the factor reversal test. (3 Marks)

Econ 245

Summer A01 2017

Page 13

.

802796.2032609.1714286.2

794118.234

95

00

00

fails

QP

qp

qp

tt

ii

itit

Question 11: Eviews (4 Marks)

Assume the data is a population. (i) What is the standard deviation of test 1?

squroot(749.75/12)=7.904376

(ii) What is the skew of test 2 using the simple Pearson measure?

[28.16667-27.75]/10.61119=0.03927

(iii) How many observations are in either series? N=12

(iv) What is the formula for “Sum Sq. Dev.”? 2

X

Econ 245

Summer A01 2017

Page 14 Formulae

Central Location: Arithmetic mean 1

Nxi

(Grouped data

x f

f Nx f

i i

i

i i

1 )

Weighted arithmetic mean W i i ix w w /

Geometric mean G ix N 1

Harmonic mean H

iN x

1 11

Dispersion: Population variance 2 21

Nxi

(Mean squared deviation)

(Grouped data 2 21

Nx fi i )

Sheppard's correction c h2 2 2 12

Sample variance sn

x xi

2 21

1

( )

Mean absolute deviation MAD = 1

Nxi

Coefficient of variation CV = ( / ) 100

Percentiles: P N Kk /100

Other Measures: Skewness coefficient skew = ( - median) /

Price Indices: Laspeyres' PL

0t = [ pit qi0 ] / [ pi0 qi0 ]

Paasche's PP

0t = [ pit qit ] / [ pi0 qit ]

Fisher's "ideal" PF

0t = [ PP

0t PL

0t ]1/2

M.E. Price Index:

P

pq q

pq q

t

ME

it

i it

i

i it

0

0

0

0

2

2

M.E. Quantity Index

Q

qp p

qp p

t

ME

it

i it

i

i it

0

0

0

0

2

2

Econ 245

Summer A01 2017

Page 15

Quantity Indices: Laspeyres' QL

0t = [ qit pi0 ] / [ qi0 pi0 ]

Paasche QP

0t = [ qit pit] / [ qi0 pit ]

Fisher's "ideal" QF

0t = [ QP

0t QL

0t ]1/2

Time Reversal test:( * )P Pt t0 0 1 Factor Reversal Test: P Qp q

p q

i i

i i

01 01

1 1

0 0

*