Dr. V. Reddy Dondeti, Associate Professor
School of Business – Norfolk State University
BUS-270 Business Statistics
Unit#2 – Review Questions
Part I – Mean and Variance
True/False or Multiple-choice questions (Answer all questions)
1. For any set of data, the range (defined as the difference between the maximum and minimum)
will always be positive. (True or False?)
2. If two datasets (or distributions) have the same range, then their standard deviations
will also be the same. (True or False?)
3. If two datasets (or distributions) have the same range, then their standard deviations will never be
the same. (True or False?)
4. The variance of a dataset or distribution is always greater than its standard deviation. (True or
False?)
5. The variance of a dataset or distribution can sometimes be negative. (True or False?)
6. The variance of a dataset or distribution can never be zero. (True or False?)
7. The variance of a dataset or distribution is always positive. (True or False?)
8. The mean and standard deviation of a dataset or distribution can never have the same value. (True
or False?)
9. For any dataset or distribution, the sum of all the deviations from the mean must always be zero.
(True or False?)
10. A machine was set up to produce steel bars with a length of 32 inches. In a sample of 100 bars,
each bar was found to be exactly 32 inches long. Then the range, variance, and standard
deviation of the sample are all equal to zero. (True or False?)
11. In a frequency distribution table, the class mid-point plays no role in the calculation of the
mean. (True or False?)
12. The variance of a dataset or distribution is always less than its standard deviation. (True or False?)
13. The formula for calculating Variance is given by: Var = SSD/((n-1) if n less than 30. (True or
False?)
14. The formula for calculating Variance is given by: Var = SSD/n if n greater than or equal to30.
(True or False?)
15. When a dataset has only a few values (less than 30), it is better to organize them in a frequency
distribution table before calculating the mean and variance. (True or False?)
16. The range is a better measure of dispersion of data than the variance (True or False?)
17. When the values in a large dataset are organized in a frequency distribution table, using relative
frequencies (or proportions), denoted by r, to calculate the mean and variance will provide you with
more information (True or False?).
18. When the values in a large dataset are organized in a frequency distribution table, using class
frequencies, denoted by f or w or g, to calculate the mean and variance will provide you with more
information (True or False?).
19. When the values in a large dataset are organized in a frequency distribution table, using relative
frequencies (or proportions), denoted by r, to calculate the mean and variance will provide you with
limited information (True or False?).
20. When the values in a large dataset are organized in a frequency distribution table, using class
frequencies, denoted by f or w or g, to calculate the mean and variance will provide you with limited
information (True or False?).
21. When the values in a large dataset are organized in a frequency distribution table, using relative
frequencies (or proportions), denoted by r, you can calculate the mean and variance faster. (True or
False?).
22. When the values in a large dataset are organized in a frequency distribution table, using class
frequencies, denoted by f or w or g, you can calculate the mean and variance faster. (True or False?).
23. If a dataset has millions of values, it is faster and simpler to use the three-column format to
calculate the mean and variance (True or False?).
24. When the values in large datasets are organized in a frequency distribution table, the sum of all
the deviations from the mean will never be equal to zero. (True or False?).
25. When the values in large datasets are organized in a frequency distribution table, you can
sometimes find the mean and variance without calculating the class mid-point. (True or False?).
26.In a frequency distribution table, the class mid-point is equal to the average of the lower boundary
and upper boundary of the class interval. (True or False?).
27. In a frequency distribution table, after the class mid-points are calculated, the class boundaries can
be ignored in the calculation of the mean or variance. (True or False?).
28. When the values in a large dataset are organized in a frequency distribution table, if you are
interested in finding the class totals (or group totals) and the grand total, you must use class
frequencies, denoted by f or w or g. (True or False?).
29. When the values in a large dataset are organized in a frequency distribution table, if you are
interested in finding the class totals (or group totals) and the grand total, you must use relative
frequencies, denoted by r. (True or False?).
30. If all the values in a dataset are either zero or one, then the mean and variance will be between
zero and one. (True or False?).
31. In a Frequency Distribution Table, the number of classes c plays no role in the calculation of the
mean or variance. (True or False?).
32. In a Frequency Distribution Table, the total of all class frequencies (i.e., the total number of values
in the dataset) represented by n plays no role in the calculation of the mean or variance. (True or
False?).
33. Class relative frequency and class proportion are the same and are usually represented by
decimal fractions, and when we multiply them one by 100, we get the corresponding percentage
value. (True or False?).
34. The concept of “class relative frequency” applies only to a class in a Frequency Distribution
Table, whereas the concept of “proportion” may apply to any group of objects with a specific
characteristic or attribute in a dataset. . (True or False?).
35. In all calculations, there is a need to convert all percentages to decimal fractions. . (True or
False?).
36. If the mean of a dataset is zero, its variance will also be zero. . (True or False?).
Part II – Probability, Discrete Distributions and Long-term Average
True/False or Multiple-choice questions (Answer all questions)
1. A sample space is the set of all possible events or outcomes associated with an experiment.
(True or False?)
2. The sum of the probabilities of all possible outcomes in an experiment is always equal to 1.
(True or False?)
3. The complement of an event includes all the events in the sample space except the event itself.
(True or False?)
4. If E is an event and EC is its complement, then, P(E) + P(EC) = 1 (True or False?)
5. If the probability P(E) of an event E is given, you can immediately calculate the probability of its
complement, P(EC). (True or False?)
6. A set of events is said to be mutually exclusive, if the occurrence of one event precludes or
eliminates the occurrence of other events in the set. (True or False?)
7. A set of events is said to be collectively exhaustive, if no other event, except the
ones in the set, can occur in an experiment. (True or False?)
8. Two events are said to be independent, if the occurrence of one event precludes or eliminates
the occurrence of the other. (True or False?)
9. Two events are said to be independent, if the occurrence of one event has no effect whatsoever on
the occurrence of the other. (True or False?)
10. If two events A and B belong to a mutually exclusive set of events, then P(A or B) = P(A)*P(B).
(True or False?)
11. If two events A and B belong to a mutually exclusive set of events, then P(A or B) = P(A)+P(B).
(True or False?)
12. If two events are independent, then P(A and B) = P(A) + P(B). (True or False?)
13. If two events are independent, then P(A and B) = P(A)*P(B). (True or False?)
14. Suppose you roll a die once. What is the probability of getting a number greater than 4?
(a) 1/3 (b) 1/2 (c) 2/3 (d) 5/6 (e) None of the above
15. When you roll a die, the probability of getting an even number is the same as getting an odd
number. ((True or False?)
16. When you a throw a dart, suppose that the probability of hitting the target is 0.40. You are
given two darts. What is the probability that you will hit the target both the times?
(a) 0.36 (b) 0.40 (c) 0.20 (d) 0.16 (e) None of the above
17. When you a throw a dart, suppose that the probability of hitting the target is 0.40. You are
given two darts. What is the probability that you will miss the target both the times?
(a) 0.36 (b) 0.40 (c) 0.20 (d) 0.16 (e) None of the above
18. In empirical studies, relative frequencies are considered equivalent to probabilities of events
or classes (True or False)
19. Surveys or experiments are used to collect data which in turn can be used to calculate relative
frequencies or proportions or probabilities.
20. Frequency distribution tables and cross-classification tables can be used to estimate
probabilities of events empirically.
——————————————————————————————–
————————————————————————————————–Refer to the following Frequency table and answer questions 21, and 22.
The relative frequencies of steel bars produced by a machine in each of the five bins or groups
classes are given in the table below along with the dimensions
Class
Number
Class Interval
Lower
Upper
Boundary
Boundary
1
2
3
4
5
20
30
40
50
60
30
40
50
60
70
Total
Proportion or
Relative
Frequency
Cumulative Relative
Frequency
0.1262
0.1790
0.4060
0.1735
0.1152
1.000
0.1262
0.1262+0.1790=0.3052
0.3052+0.4060=0.7112
0.7112+0.1735=0.8848
0.8848+0.1152=1.00
(21) What is the probability that a steel bar selected at random has a length of less than or equal
to 50 inches? (Hint: Calculate the cumulative relative frequency of class 3)
(a) 0.519
(b) 0.711
(c).892 (d) 0.612
(e) 0.437
(22) What is the probability that a steel bar selected at random has a length greater than or equal
to 50 inches?
(a) 0.289
(b) 0.372
(c) 0.646 (d) 0.573
(e) 0.422
Hint: Add the relative frequencies of classes 4 and 5: 0.1735+0.1152 = 0.2887 or 0.289
or Use the rule: P(EC) = 1─ P(E). In this case, P(X≥50) = 1─P(X≤ 50) = 1 − 0.711 = 0.289
Refer to the following cross-classification table and answer questions 23, 24, 25, and 26.
In a survey by a new car dealer, a total of 1000 men and women were asked about the sizes of
cars they prefer to drive. The number of people that preferred full-size, mid-size, or compact-size
cars are given below in the Table. Find the missing values and calculate the probabilities based
on the empirical data.
Men
Women
Full-size car
147
?
240
Mid-size car
?
248
?
Compact-size car
?
186
310
Total
?
?
1000
23. What is the probability that a person selected at random would prefer to drive a full-size car?
(a) 0.15 (b) 0.93 (c) 0.24 (d) 0.49 (e) None of the above
24. What is the probability that a person selected at random would prefer to drive a mid-size car?
(a) 0.45 (b) 0.20 (c) 0.64 (d) 0.25 (e) None of the above
25. What is the probability that a woman selected at random would prefer to drive a full-size car
or compact-size car?
a) 0.371 (b) 0.279 (c) 0.312 (d) 0.238 (e) None of the above
26. What is the probability that a man selected at random would prefer to drive a full-size car or
mid-size car?
(a) 0.271 (b) 0.486 (c) 0.324 (d) 0.452 (e) None of the above
Solutions to questions 23, 24, 25 and 26 are given below:
Step 1: Number of women who preferred a full-size car = 240 – 147 = 93
Step 2: Number of men who preferred a compact-size car = 310 – 186 = 124
Step 3: Total number of people who preferred mid-size car = 1000 – 240 – 310 = 450
Step 4: Number of men who preferred a mid-size car = 450 – 248 = 202
Step 5: Total number of men in the sample = 147 + 202+ 124 = 473
Step 6: Total number of women in the sample = 1000 – 473 = 527
Men
Women
Full-size car
147
93
240
Mid-size car
202
248
450
Compact-size car
124
186
310
Total
473
527
1000
23. What is the probability that a person selected at random would prefer to drive a full-size car?
(a) 0.15 (b) 0.93 (c) 0.24 (d) 0.49 (e) None of the above
(Answer: 240/1000=0.24)
24. What is the probability that a person selected at random would prefer to drive a mid-size car?
(a) 0.45 (b) 0.20 (c) 0.64 (d) 0.25 (e) None of the above
(Answer: 450/1000=0.45)
25. What is the probability that a woman selected at random would prefer to drive a full-size car
or compact-size car?
(a) 0.371 (b) 0.279 (c) 0.312 (d) 0.238 (e) None of the above
(Answer: (93+186)/1000=0.279)
26. What is the probability that a man selected at random would prefer to drive a full-size car or
mid-size car?
(a) 0.271 (b) 0.486 (c) 0.324 (d) 0.452 (e) None of the above
(Answer: (147+124)/1000=0.271)
Other questions can be answered in a similar fashion.
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