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Frequency tables
Exercise 1:
Given the following stem-and-leaf display:
9
714
10
82230
11
561776735
12
394282
13
20
1. Discuss the shape of the distribution of this data set.
2. Rearrange the leaves and form the revised stem-and-leaf display
3. Place the data into an ordered array
4. Which of the two devices seems to give more information? Discuss
Exercise 2:
Upon examining the monthly billing records of a mail-order book company, the auditor
takes a sample of 20 of its unpaid accounts, The amounts owed to the company are:
$
4 $ 18 $
11 $
7 $
7 $
10 $
5 $
33 $
9 $
12
$
3 $
10 $
6 $
26 $
37 $
15 $
18 $
10 $
21
11 $
1. develop and ordered array
2. form the stem and leaf display as well as the revised display.
3. Discuss the skewedness of the distribution of this data set.
Exercise 3:
The Struge rule recommends using as number of classes CS=1+3.3log10 (n) where n is the
sample size and log10 is the logarithm of base 10 function. (log10 (n)=x is equivalent to
10x =n). The square root rule recommends however using CR= n .
Using these sample sizes, compute CS and CR.
5,10,20,100,2000,69000,500000,32000000
What rule would imply in each case? Why?
Exercise 4:
What follows is a sample of 20 daily revenues for the Caldense bakery:
$108
$126
$36
$241
$120
$21
$30
$25
$19
$93
$25
$265
$10
$69
$66
$79
$89
$96
$75
$48
1. develop the ordered array
2. form the stem-and-leaf display
3. form the revised display
4. do you distinguish any outlier in the previous data
Summary Exercise: frequency tables and histograms
The following data are the retail prices of a random sample of 30 pencil tire pressure
gauge models
4.5
6.5
2
2.5
4
3.5
5
3
5
5.5
1
7.5
3
2
3
3.5
5
3.5
6
4.5
2
3.5
3.5
3
3
3
4
1.5
2.5
1.5
(a) Place the raw data in the stem-and-leaf display
(b) Place the raw data in an ordered array
(c) Form the frequency distribution and the relative frequency distribution
(d) Plot the relative frequency histogram
(e) Plot the percentage (relative frequency) polygon
(f) Form the cumulative relative frequency distribution
(g) Plot the ogive (cumulative frequency polygon)
(h) If you were considering buying a pencil tire pressure gauge, what would you want to
know else.
Measures of central tendancy
Exercise 1:
Two basketball players scored respectively these totals of points in their respective
playoff games.
Game
Karim
Jim
1
27
17
2
22
19
3
23
21
4
24
21
5
25
24
6
5
21
7
3
19
8
13
16
9
19
22
10
23
19
11
32
24
(a) Compute major central tendency measure for the previous data
(b) The team manager would like to trade one of the 2 players, who would you
recommend to keep?
(c) In view of the fact that a severe flue during game 6 and 7 that prevented him from
playing at his best level struck Karim, would you provide a different
recommendation? Justify your answer numerically.
(d) Discuss the differences in concept between the mean and the median and how this
relates to (b) and (c).
Exercise 2:
The following table depicts the frequencies recorded for the grades (the maximum grade
is 20).
Class of grades
midpoint
Frequency
0-2
2-6
6-10
5
relative
frequency
10%
13
cumulative
frequency
cumulative relative
frequency
65%
75%
1. Fill the missing entries in the following frequency table.
2. What is the size of the sample here?
3. Compute the mean for the previous distribution of grades
4. Compute the standard deviation
Exercise 3:
The following distribution gives the dollar cost per unit of output for 200 plants in the
same industry:
Dollar Cost per
Unit of Output
$1.00 and under $1.02
1.02 and under 1.04
1.04 and under 1.06
1.06 and under 1.08
1.08 and under 1.10
1.10 and under 1.12
1.12 and under 1.14
1.14 and under 1.16
Total
Number of Plants
6
26
52
58
39
15
5
1
200
(a) Calculate the arithmetic mean of the distribution.
(b) Calculate the median of the distribution.
(c) Can you say that 50% of the units produced cost less than the median calculated in
part (b)? Explain.
Suppose that the last class had been “$1.14 and over”. What effect, if any, would this
have had on your calculation of the arithmetic mean and the median? Justify your answer.
Exercice 4
For the period 1981-1990, the annual earnings per share for two companies are as follow:
Year
A
B
1981
0.50
6.4
1982
0.8
7
1983
0.9
6.8
1984
1.2
7.6
1985
1
8
1986
0.8
8.3
1987
1.2
7.9
1988
1.4
8.5
1989
1.5
8.6
1990
1.7
8.9
a- Compute the arithmetic mean and standard deviation of the earnings per share for
each firm. Which firm showed the greater absolute variation in earnings per share?
b- Compute the coefficient of variation for each firm. Which firm showed relatively
greater variation in earnings per share?
Set 3 of exercises on data Analysis
Exercise 1:
Jordi and Amy Fernandez have a house, a dog, and three beautiful children: Steven 9
years, Miguel 6 years and Santina 3 years.
(a) Knowing that the average age of the family is 17.4 and its standard deviation is 15,
what are the respective ages of Jordi and Amy (Assuming that Jordi is Older).
(b) What would be the average and standard deviation of ages six years later?
(c) Same question as (b) assuming that Jordi and Amy have a baby in three years from
now?
(d) A mean of 25 and a standard deviation of 16 characterize the Rogers family’s ages.
What would be the average and standard deviation of the ages of both families.
(assuming that both households together add up to 11 individuals)
Exercise 2:
The caldense bakery has recorded the following sales for the first 10 days of September.
$25
$25
$96
$48
$21
$108
$265
$75
$69
$66
1- determine the following central tendency measures
Mean=
Q1=
IQ range=
Range=
Median=
Q3=
st-dev=
Midrange=
geo mean=
top 20%=
variance=
2- The plot of the cumulative relative of the 20 first days in September is the following
cum. Freq
cum. Freq Caldense
100%
90%
80%
cumulative frequency
70%
60%
50%
40%
30%
20%
10%
0%
-
40
80
120
160
200
240
280
daily revenue
a- Retrieve the quartiles from the above graph and show in the graph the interquartile
range.
b- From the graph above, what is the proportion of sales less than 60%? Less than
100%? Where are located the top10% sales?
c- Locate in the graph the inner fences and the outer fences. Do you detect any outliers?
3- The actual data for the 20 first days in September are the following
$
$
$
$
25
25
96
48
$
$
$
$
69
66
74
26
$
$
$
$
30
81
99
159
$
$
$
$
21
108
265
75
$
$
$
$
5
201
66
77
Reconstitute the cumulative relative frequency. Answer the same question as in 2- from
the raw data.
Exercise 3
x,y and z are statistical variables and a,b, and c are positive constants. Prove that:
•
Var(ax+by)=(aσx -bσy )2
•
Var(a)=0
•
Cov(x,x)=var(x)
•
Cov(ax+y,z)=aCov(x,z)+Cov(y,z)
•
Cov(a,x)=0
•
ρax,by+c=ρx,y