PRACTICE EXAM 1 Business Statistics I
Examination I, Chapters 1-3
Payne
Part 1. Problem Solving. Please do all work on your own paper; do not write your work on this test form. 75 points (includes 5
bonus); point values are in [ ]. For each problem:
•
Write the formula, where applicable.
•
Set up the problem IN SYMBOLS and show all calculations. SHOW ALL WORK.
•
Answer the problem. You may round to ≥3 decimals. Please circle your answers.
YOU MUST ARRANGE YOUR WORK IN LOGICAL ORDER TO RECEIVE CREDIT!
1.
Consider the following hypothetical sample of 6 Statistics students. COLLSEM is the number of semesters of college completed.
(i)
(x)
ID COLLSEM
Here is the solution to part d, the Skewness coefficient.
1
5
3
⎛ x − x⎞
2
3
⎟
⎜
3
n
3
2
⎛ x − x⎞
⎝ s ⎠
n
x− x
ID COLLSEM
⎜
⎟
Sk =
4
6
(n − 1)(n − 2) i = 1 ⎜⎝ s ⎟⎠
1
5
0.3333
0.0057
5
5
2
3 -1.6667 -0.7173
6
6
7
=
( −13083
.
)
3
2 -2.6667 -2.9379
∑
4
5
6
6
5
7
28
1.3333
0.3333
2.3333
0.3672
0.0057
1.9682
-1.3083
(6 − 1)(6 − 2)
FOR THE VARIABLE COLLSEM:
[10] a. Calculate the coefficient of
= -0.392
variation (CV).
[6]
b. Find P70.
[4]
c. What is the mode?
Interpretation: The data show a slight negative skewness.
[x]
d. What is the skewness
coefficient? Interpret this result. (NEW)
2.
A hypothetical sample of public school administrators provided information on the dollar value of their contributions to the Democratic
Party for 2004. The data were summarized in the following frequency distribution.
Class
CONTRIB
1 $0 up to 250
2 $250 up to 500
3 $500 up to 750
4 $750 up to 1000
SUMS
f
4
7
5
12
CF m (midpt)
4
124.5
11
374.5
16
624.5
28
874.5
f*m
498.00
2,621.50
3,122.50
10,494.00
$16,736.00
m^2
15,500.25
140,250.25
390,000.25
764,750.25
f*m^2
62,001.00
981,751.75
1,950,001.25
9,177,003.00
$12,170,757.00
FOR THE VARIABLE CONTRIB:
[4]
a. What is the value of n, the number of observations?
[6]
b. Estimate the sample mean contribution.
[5]
c. Estimate the sample standard deviation.
[4]
d. What is the relative frequency of class 3 ($500-$750)?
3. [8] Bob completed 5 courses at Calhoun last semester. Using the data shown below, and using credit hours as the weight variable
and quality points as the x variable, calculate his grade point average (GPA) for the semester as a weighted arithmetic mean.
Course
Western Civ II
Economics I
Biology I
College Algebra
English Lit I
4.
Grade
B
A
B
C
A
Quality points
3
4
3
2
4
Credit hours
3
3
4
4
3
[10] Consider these inflation rates for 5 selected western European countries:
1.5%
0.4%
2.4%
1.5%
0.3%
Calculate the geometric mean inflation rate.
5.
[8] Consider these parameters for data set x:
Calculate the population variance of x.
6.
[6] Refer to the box-and-whisker plot below.
a. What is the (approximate) value of Q3, the third quartile?
b. Does the distribution appear to be negatively skewed, positively skewed, or symmetrical?
10
0
7.
[4]
20
30
40
50
60
70
80
90
Consider the following 4 frequency polygons. Which best represents a skewness coefficient of +1.95?
f
f
f
x
x
Diagram A
f
Diagram
x
Diagram
x
Diagram D
-----END OF PART 1
Part 2. True or False. Write a T or F on your own paper. 1 point each, 10 points total.
1.
Carrie observes the dogs at a dog show and organizes the data into a frequency distribution, classifying the observations by
breed of dog. This is an example of ordinal data.
2. An example of ratio level data would be the gasoline consumption per mile of a set of automobiles.
3. Barbara studies a distribution of variable R which contains 1,000 observations. According to Chebyshev’s Theorem, at least 850
of the observations must lie within 2 standard deviations of the mean.
4. A statistic measures some characteristic of a sample.
5. The median of a group of data is the value which represents the center of the "weight" of the distribution, or the "balance
point."
6. For a given distribution, mode < median < mean. The distribution is positively skewed.
7. The standard deviation shows the average difference between the observations in a data set and the mean of that data set.
8. A parameter is a quantity that has a variable definition and a variable value.
9. Say that you wish to compare the dispersion between two data sets. Variable x consists of the weights of a sample of Mack
trucks. Variable y consists of the number of hours required to make each of the trucks. Of the measures of dispersion we
have studied, the best one to make the comparison is the coefficient of variation (CV).
10.
Say that you study a variable, T. You find that, no matter what two possible values you select, an infinity of other
possible values will lie between them. The variable T must be a continuous variable.
11.
John completes a study which shows that heavy use (2 bottles or more per day) of hand lotion made with radish pulp
decreases the risk of skin cancer by 50%. Women around the country begin using large quantities of radish-pulp hand
lotion, but with no effects on skin cancer rates. John’s study apparently lacks external validity.
12.
Say that a properly designed and properly executed sample of the shrimp population living off the coast of Crab Island
finds a mean weight of 2.143 ounces. The true population mean is actually 2.259 ounces. The difference is sampling
error.
13.
A sample can be more accurate than a census only if the sample is itself very large.
END OF IN-CLASS PORTION OF EXAM ******Please submit your homework before you leave. (10 points)
Part 3. [10] Frequency distribution and graph construction. DUE: BEGINNING OF THE NEXT CLASS. Answers
may vary.
DATA: School life expectancy (expected number of years of formal schooling), from a sample of 54 countries, most
recent available data. (Source: Statistics Division, United Nations.
http://unstats.un.org/unsd/demographic/products/socind/education.htm)
11
11
16
11
13
20
15
10
13
8
14
14
19
7
15
7
14
12
15
15
13
12
3
5
9
9
16
12
5
14
10
14
15
11
7
8
11
6
13
13
13
15
17
4
12
12
11
9
5
16
5
18
15
12
A. FREQUENCY DISTRIBUTION: Construct a frequency distribution with the following characteristics:
a.
A "reasonable" number of classes. Describe briefly how you arrived at this number. Mathematically defensible
methods are preferred to "ad hoc" methods.
b.
Give the frequency, relative frequency, cumulative frequency, and relative cumulative frequency for each class.
(I.e., a standard complete frequency distribution)
c.
Make a complete stem and leaf diagram for these data
B. GRAPH:
i.
ii.
iii.
iv.
v.
vi.
Draw any one of the following graphs to represent the data: Use pencil and paper, not a computer.
histogram
frequency polygon
less-than-cumulative frequency ogive
more-than-cumulative frequency ogive
pie graph
box-and-whisker plot
Briefly explain (a) why you chose to use this type of graph and (b) what it shows about the data and the datagenerating process.