Midterm 1

Question 1

What do measures of central tendency yield?

Answer 1

Measures of central tendency yield information about the center (middle) of a group of numbers

Question 2

What is the mode?

Answer 2

Mode: the most frequently occurring value in a data set
* Applicable to all levels of data measurement
* Sometimes no mode exists or there is more than one mode (bimodal or multimodal)
* Often used with nominal/ordinal data (e.g., determining the most common hair color/ blood type)

Question 3

What is the median? What are some advantages and disadvantages?

Answer 3

Median: the middle value in an ordered array of numbers
* Array values in order
* The median of the array is the center number, or with an even number of observations, the average of the middle two terms
* Advantage: not affected by extreme values, so often preferable to the mean when the data includes some unusually large or small observations (e.g., income in the U.S., house prices in a given area)
* Disadvantage: it does not include all of the information in the data
* Data measurement level must at least be ordinal

Question 4

What is the arithmetic mean?

Answer 4

Arithmetic Mean: the average of a group of numbers
* Most common measure of central tendency
* Includes all information in the data set

Question 5

What are percentiles?

Answer 5

Percentiles: measures of central tendency that divide a group of data into 100 parts
* At least n% of the data lie at or below the nth percentile, and at most (100 - n)% of the
data lie above the nth percentile
* Example: 90th percentile indicates that at least 90% of the data are equal to or less
than it, and 10% of the data lie above it

Question 6

What are quartiles?

Answer 6

Quartiles: measures of central tendency that divide a group of data into four subgroups
25% of the data set is below the first quartile
50% of the data set is below the second quartile (also called the median) 75% of the data set is below the third quartile
100% of the data set is below the fourth quartile

Question 7

What are measure’s of variability

Answer 7

Measures of variability: describe the spread or dispersion of a set of data
* Distributions may have the same mean but different variability

Question 8

Explain range and give an advantage and a disadvantage

Answer 8

Range: the difference between the largest and the smallest values in a set of data
* Advantage – easy to compute
* Disadvantage – affected by extreme values

Question 9

What is interquartile range?

Answer 9

Interquartile range: range of values between the first and third quartile
* Range of the “middle half”; middle 50%
* Useful when analysts are interested in the
middle 50% and not the extremes

Question 10

What is variance?

Answer 10

Variance: average of the squared deviations about the arithmetic mean for a set of numbers

Question 11

What is the Standard Deviation and what does it allow for?

Answer 11

Standard Deviation: square root of the variance
* Closely related to the variance but more easily interpretable
* The standard deviation allows us to apply the empirical rule and Chebyshev’s Theorem

Question 12

Explain the empirical rule

Answer 12

Used to state the approximate percentage of values that lie within a given number of standard deviations from the mean of a set of data if the data are normally distributed
* Data must be normally distributed
* Since this is common for many things, the empirical rule is widely used

Question 13

Explain Chebyshev’s Theorem

Answer 13

Chebyshev’s theorem tells us at least what percentage of the data will lie within a certain range; if the distribution is closer to normal, the actual amount will be greater

Unlike the empirical rule, data can have any distribution

For example, 75% of data will lie within 2 standard deviations of the data, no matter how the data is distributed

Question 14

Sample variance and standard deviation estimate what? Why is the denominator important?

Answer 14

The sample variance and standard deviation are used as estimators of the population values

The denominator is (n − 1) rather than N, which makes the sample statistics unbiased estimators of the population parameters

Question 15

Explain z-scores and what the z scores represent if positive or negative

Answer 15

z Scores: represent the number of standard deviations a value (x) is above or below the mean for normally distributed data

Negative z scores indicate that the raw value (x) is below the mean; positive z scores indicate x values above the mean

Question 16

What is the coefficient of variation? What does it measure?

Answer 16

The Coefficient of Variation: ratio of the standard deviation to the mean expressed as a
percentage

The CV can be used as a measure of risk

Question 17

What are measure’s of shape?

Answer 17

tools that can be used to describe the shape of a distribution of data

Question 18

What is skewness?

Answer 18

is when a distribution is asymmetrical or lacks symmetry

Skewed portion is the long, thin part of the curve

Question 19

Explain in depth how measure’s of central tendency relate to skewness

Answer 19

• The relationship of the mean, median, and the mode relate to skew
• Symmetric: mean, median, and mode are equal
• Negatively skewed: mean is less than the median, which is less than the mode
• Positively skewed: mode is less than the median, which is less than the mean

Question 20

What does Kurtosis describe?

Answer 20

the amount of peakedness of a distribution

Question 21

Explain the box-and-whisker plot

Answer 21

a diagram that utilizes the upper and lower quartiles along with the median and the two most extreme values to depict a distribution graphically
Sometimes called the 5-number summary
* A box is drawn around the median with the upper and lower quartiles as the box endpoints
(hinges)
* The interquartile range is used to construct the inner fences, ± 1.5 ∙ IQR
* If data fall outside the inner fences, outer fences are constructed, ± 3.0 ∙ IQR
* A line segment (whisker) is drawn from the lower hinge of the box outward to the smallest data value
* A second whisker is drawn from the upper hinge to the largest data value

Question 22

What is the use of box and whisker plots?

Answer 22

One use of box-and-whisker plots is to find outliers
* Data values that fall outside the mainstream of values in a distribution are called outliers
o Sometimes merely extremes of the data
o Sometimes due to measurement or recording error
o Sometimes so unusual that they should not be considered with the rest of the data
* Values that are outside the inner fences but inside the outer fences are mild outliers
* Values that fall outside the outer fences are extreme outliers

Another use is to determine if the distribution is skewed
* The position of the median in the box gives information about the skew of the middle 50% of the data
o If the median is to the left, the middle 50% is skewed right
o If the median is to the right, the middle 50% is skewed left
* The length of the whiskers shows the skew of the outer values

Question 23

Why do business analytics use descriptive statistics?

Answer 23

• Descriptive statistics are at the foundation of statistical techniques and numerical measures that can be used to gain an initial understanding of data in business analytics
• Descriptive statistics allows a business analyst begin to mine and understand any meanings and/or relationships that might exist in data

Question 24

What is a (random) experiment? Give an example

Answer 24

a process that produces well-defined outcome(s)
Sampling every 200th bottle of cola and weighing it

Question 25

What is an event, give an example

Answer 25

an outcome of an experiment

There are 10 bottles that are too full

Question 26

What is an elementary event? Give an example

Answer 26

event that cannot be decomposed or broken down into other events

o Elementary events are denoted by lowercase letters

o Suppose that the experiment is to roll a die
o Elementary events are to roll a 1, a 2, a 3, etc.
o In this case, there are six elementary events, e1, e2, etc.

Question 27

What is the sample space?

Answer 27

a complete listing of all elementary events (all possible outcomes ) for a random experiment

Question 28

What is the classical method of assigning probability?

Answer 28

The probability of an individual event occurring is determined by the ratio of the number of items in a population that contain the event (ne) to the total number of items in the population (N)

• Because ne can never be greater than N, the highest value of a probability is 1
• The lowest probability, if none of the N possibilities has the desired characteristic, e, is 0
• Thus, 0≤P(E)≤1

Question 29

What is a priori probability?

Answer 29

(classical probability)– the probability can be
determined before the experiment takes place

Question 30

What is the relative frequency of occurrence (empirical probability)?

Answer 30

Probability of an event occurring is equal to the number of times the event has occurred in the past divided by the total number of opportunities for the event to have occurred

Based on historical data; the past may or may not be a good predictor of the future

Question 31

What is subjective probability? Give an example

Answer 31

• Based on the insights or feelings of the person determining the probability
• Different individuals may (correctly or incorrectly) assign different numeric probabilities to the same event
• subjective approach is usually limited to experiments that are unrepeatable

An experienced airline mechanic estimates the probability that a
particular plane will have a certain type of defect

Question 32

Explain the Venn diagram structure of probability

Answer 32

• Rectangular area represents the sample space for the random experiment and contains all possible outcomes.
• Circle represents event A and contains only the outcomes that belong to A.
• Shaded region of the rectangle contains all outcomes not in event A.

Question 33

What is a mutually exclusive event?

Answer 33

• Mutually Exclusive Events
  o Events with no common outcomes
  o Occurrence of one event precludes the occurrence of the other event
  o Example: if you toss a coin and get heads, you cannot get tails

Question 34

What are collectively exhaustive events?

Answer 34

o Contains all possible elementary events for an experiment
* Rolling a die: {1,2,3,4,5,6}
* Generating a random integer number: { >5,= <5}
o The sample space for an experiment can be described as mutually exclusive (events do not have any outcome in common) and collectively exhaustive

Question 35

What are complementary events?

Answer 35

o Given an event X, the complement of X is defined to be the event consisting of all outcomes that are not in X.
o Complementary events are denoted X′ (or 𝑋􏰊), which is pronounced as “not X”
o In any probability application, either event X or its complement X′ must occur.
P(X′) = 1 − P(X)

Question 36

Explain unions and intersections

Answer 36

• Set notation is the use of braces to group numbers o The union of sets X, Y is denoted X ∪Y
• Given two events X and Y, the union of X and Y is defined as the event containing all outcomes belonging to X or Y or both
  o The intersection of sets X, Y is denoted X ∩ Y
• An element is part of the intersection if it is in set X and set Y

Question 37

Describe Addition Laws, when does a special case arise?

Answer 37

The General Law of Addition (addition law) is used to find the probability of the union of two events
* The probability that event A or event B or both will occur (at least one of two events will occur).
P ( X ∪ Y ) = P ( X ) + P (Y ) − P ( X ∩ Y )

A special case arises for mutually exclusive events.

Question 38

Under addition laws, what is a probability matrix, joint probabilities, and marginal probabilities?

Answer 38

A probability matrix displays the intersection (joint) probabilities along with the marginal probabilities of a given problem
* When values give the probability of the intersection of two events, the probabilities are called joint probabilities.
o Inner cells show joint probabilities
* Marginal probabilities are found by summing the joint probabilities in the corresponding row or column of the joint probability table.
o Outer cells show marginal probabilities

Question 39

What is the Counting rule? give an example

Answer 39

The mn Counting Rule:
* If an operation can be done m ways and a second operation can be done n ways, then there are mn ways for the two operations to occur in order
o A cafeteria offers 5 salads, 4 meats, 8 vegetables, 3 breads, 4 desserts, and 3 drinks
* How many meals are available?
* 5 × 4 × 8 × 3 × 4 × 3 = 5760

Question 40

Explain sampling from a population with replacement

Answer 40

Sampling from a Population with Replacement:
* Sampling n items from a population of size N begin underline with replacement end underline would provide (N) n possibilities
o Six lottery numbers are drawn from the digits 0 to 9, with replacement

Question 41

Explain sampling from a population without replacement

Answer 41

Sampling n items from a population of size N without replacement provides the following number of possibilities

Question 42

What are independent events?

Answer 42

o The occurrence or nonoccurrence of one event does not affect the occurrence or nonoccurrence of the other event(s)
o The probability of someone wearing glasses is unlikely to affect the probability that the person likes milk
o Many events are not independent
* The probability of carrying an umbrella changes when the weather
forecast predicts rain If events are independent, then:
P ( X |Y ) = P ( X ), and P (Y | X ) = P (Y )
P(X |Y ) is the probability that X occursbegin underline given thatend underline Y has occurred.

Question 43

What is conditional probability?

Answer 43

Conditional probability: When the probability of one event is dependent on whether some related event has already occurred.

Conditional probabilities can be computed as the ratio of joint probability to a marginal probability.

Question 44

What are multiplication laws?

Answer 44

General Law of Multiplication
P ( X ∩ Y ) = P ( X ) ⋅ P (Y | X ) = P (Y ) ⋅ P ( X | Y )
* Used to find the joint probability

Question 45

What is the special law of multiplication?

Answer 45

Special Law of Multiplication
* If X and Y are independent,
P(X ∩ Y) = P(X) · P(Y)

Question 46

What are independent events under conditional probability?

Answer 46

Independent Events
If events are independent, then
P ( X | Y ) = P ( X ) and P (Y | X ) = P (Y )

Question 47

Explain the law of conditional probability?

Answer 47

Law of Conditional Probability: the conditional probability of X occurring, given that Y is known or has occurred is expressed

Question 48

What is Baye’s Rule?

Answer 48

Bayes’ Rule extends the use of the law of conditional probabilities to allow revision of original probabilities with new information

o The denominator is a weighted average of the conditional probabilities, with the weights being the prior probabilities
o The formula allows statisticians to incorporate new information to revise probability estimates

Question 49

What is statistics?

Answer 49

o A science dealing with the collection, analysis, interpretation, and presentation of numerical data
o Collect data -> analyze data -> interpret data -> present findings

Question 50

Population Vs Sample

Answer 50

Population: all
 A collection of all persons, objects, or items under study
 Can be broadly or narrowly defined

Census: gathering data from the whole population

Sample: gathering data on a subset of the population
 Should be representative of the whole population
 Use information about the sample to infer about the population

Question 51

What are the two branches of statistics?

Answer 51

Descriptive
 Uses data gathered on a group to describe or reach conclusions about that same group
 Produces graphical or numerical summaries of data

Inferential
 Gathers data from a sample and uses the statistics generated to reach conclusions about the population from which the sample was taken
 Sometimes called inductive statistics

Question 52

What is a parameter?

Answer 52

Parameter: descriptive measure of the population

Question 53

What is s statistic?

Answer 53

• Statistic: descriptive measure of a sample

Question 54

What are the levels of data measurement?

Answer 54

Nominal -> ordinal -> interval -> ratio data (levels of data)

Question 55

What is nominal data?

Answer 55

Nominal: Used only to classify or categorize
 No quantitative value statement is implied
 Lowest level of measurement
Examples
* Profession (doctor, lawyer…)
* Sex (male, female)
* Eye color (brown, green, blue)
* Location (zip code)
 Best way to represent is by pie charts
 For example, the name of a class (9200) and (9201) is simply the name and there is no meaning attached to the numbers therefore it is nominal. 9201 doesn’t have more seats than 9200

Question 56

What is ordinal data?

Answer 56

Ordinal: ranking or ordering
 Distances between ranks are not always equal
 Nominal and ordinal data are nonmetric data or qualitative data because their measurements are imprecise

Example
* Ranking mutual funds by risk
* 50 most-admired companies
* Coffee cup size
 Often used in surveys
* Like a professor very much, not very much, worst professor ever

Question 57

What is interval data?

Answer 57

Interval: numerical data in which the distances between consecutive numbers have meaning
 Interval data have equal intervals

Example
* Fahrenheit temperature scale
o The zero point is a matter of convenience or convention
o A temperature of 0 does not mean that there is no temperature
o The amounts of heat between consecutive readings are the same
* Time
 0 is a value*

Question 58

What is ratio data?

Answer 58

Ratio: numerical data in which the distances between consecutive numbers have meaning and the zero value represents the absence of the characteristic being studied
 Highest level of data measurement
 Interval and ratio data are called metric or quantitative data because their measurements are precise
* Example
o Volume
o Weight
o Kelvin temperature

Question 59

Metric Vs Nonmetric data

Answer 59

o Nominal and ordinal (qualitative)
o Interval and ratio (quantitative)

Question 60

What does parametric statistics require?

Answer 60

require interval or ratio data

Question 61

What can nonparametric statistics be used with?

Answer 61

can be used with any data, but nominal and ordinal data require nonparametric methods

Question 62

What is Big data?

Answer 62

A collection of large and complex datasets from different sources that are difficult to process using traditional data management and processing applications

Question 63

What are the 5 V’s?

Answer 63

Volume
 Ever-increasing size of data and databases

Velocity
 The speed with which the data are available and can be processed

Variety
 Different forms and sources of data

Veracity
 Data quality, correctness, and accuracy

Value
 Sometimes considered a fifth characteristic

Question 64

What are the categories of Business Analytics?

Answer 64

• Descriptive analytics
• Predictive analytics
• Prescriptive analytics

Question 65

What is descriptive analytics?

Answer 65

-Descriptive analytics: takes traditional data and describes what has or is happening in a business
o Used to discover hidden relationships and patterns
o Simplest and most commonly used category
o Data visualization
o Also called reporting analytics

Question 66

What is Predictive Analytics?

Answer 66

Predictive analytics: finds relationships in the data that are not readily apparent with descriptive analytics
o Patterns or relationships are extrapolated forward in time and the past is used to make predictions about the future
o Topics include, regression, time-series, forecasting, data mining, statistical modeling, machine learning techniques, decision tree models, and neural networks

Question 67

What is Prescriptive analytics?

Answer 67

Prescriptive analytics: examines current trends and likely forecasts to make better decisions
o Optimization models are an example of prescriptive analytics
o Takes uncertainty into account, recommends ways to mitigate risks, and tries to foresee the effects of future decisions
o Uses a set of mathematical techniques that determine optimal decisions given a complex set of objects, requirements, and constraints
o Topics include management science or operations research aimed at optimizing performance of a system such as mathematical programming, simulation, and network analysis

Question 68

What is data mining?

Answer 68

Data mining: collecting, exploring, and analyzing large volumes of data to uncover hidden patterns to enhance decision making

Question 69

What is Data visualization?

Answer 69

Data visualization: the study of the visual representation of data and is employed to convey data or information by imparting it as visual objects

Question 70

What is a discrete random variable?

Answer 70

Discrete random variable
o If the set of all possible values is at most a finite or a countably infinite number of possible values
o Most of the time produce nonnegative whole numbers
o Example: A group of 6 people are randomly selected from a population and the number of left-handed people are to be determined, the random variable produced is discrete because the only possible numbers are {0,1,2,3,4,5,6}, it is impossible to obtain a non-whole number.

Question 71

What are continuous distributions?

Answer 71

Take on values at every point over a given interval
o No gaps or un-assumed values
o Are generated from things that are measured
o Examples are:
 Time, weight, height, and volume
o Once this type of data is recorded it becomes discrete data because the data is rounded off to a discrete number

Question 72

What are the 3 types of discrete distributions?

Answer 72

• Binomial distribution
• Poisson distribution
• Hypergeometric distribution

Question 73

What are the 6 continuous distributions?

Answer 73

• Uniform distribution
• Normal distribution
• Exponential distribution
• T distributions
• Chi-square distribution
• F distribution

Question 74

What are the binomial assumptions?

Answer 74

Binomial assumptions
 The experiment involves n identical trial
 Each trial has only 2 possible outcomes denoted as success or failure
 Each trial is independent of the previous trials
 The terms p and q remain constant throughout the experiment, where the term p is the probability of getting a success on any one trial and the term q = 1 – p is the probability of getting a failure on any one trial

Question 75

Binomial trials must be what?

Answer 75

Independent
o This means that either the experiment is by nature one that produces independent trials, or the experiment is conducted with replacement.

Question 76

Explain the mean and standard deviation for a binomial distribution?

Answer 76

A binomial distribution has an expected value or a long-run average, which is denoted by u and the value of  is determined by n*p

The standard deviation of a binomial distribution is denoted by SD = (square root)npq

Question 77

What is the Poisson distribution?

Answer 77

• Poisson distribution focuses only on the number of discrete occurrences over some interval or continuum
• Another discrete distribution
• Has been referred to as the law of improbable events
• Often used to describe the number of random arrivals per some time interval

Question 78

What are the characteristics of the Poisson distribution?

Answer 78

• Discrete distribution
• Describes rare events
• Each occurrence is independent of the other occurrences
• It describes discrete occurrences over a continuum or interval
• The occurrences in each interval can range from zero to infinity
• The expected number of occurrences must hold constant throughout the experiment

Question 79

Give an example of a Poisson distribution

Answer 79

• Number of telephone calls per minute at a small business
• Number of hazardous waste sites per province in Canada