EC 255 Lecture Notes PDF

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Chapter 1 Statistics: - Art and science of gathering, analyzing, interpreting, and presenting data

Variable: - Characteristic that takes different values Measurement: - Standard practice; used to assign values to a variable Data: - Measurements that are collected, recorded, and summarized for presentation, analysis, and interpretation Types of Data: Nominal: - Can’t be ranked - Ie eye color, sex Ordinal: - Can be ranked - Ie good, fair, poor Interval: - Distances between consecutive values have meaning - Numerical - 0-point is matter if convention / convenience - Ie Celsius, Fahrenheit

Ratio: - Distance between consecutive values have meaning - Numeral, NON-negative - Values can take on a natural or absolute zero - Ie Kelvin temp, height, mass, time QUANTITATIVE DATA: Discrete: - Distinct value - Whole numbers, binary Continuous: - Values within range - Can be split into infinite values - Weight can have an infinite number of decimals MORE DEFINITIONS: Population: - Collection of all possible elements of interest Census: - Collection of values for all variables of interest that correspond to all elements of a population - Size denoted by N Parameter: - Summary measure used to describe values of a variable for the entire population Sample: - Collection of elements that comprise of a subset of population - Size denoted by n Statistic: - Estimate of the value of a parameter based on the elements that belong to the sample

Descriptive vs Inferential Statistics Descriptive: - Tabular, graphical, numeral methods to summarize data - Data -> information

Inferential: - Use data obtained through sampling to estimate value of parameter or test a hypothesis about parameter (inductive logic) - Information -> knowledge

Chapter 2: Visualizing Data with Tables, Charts, and Graphs Ungrouped vs. Grouped Data

Frequency Distributions: ● a frequency distribution of a variable is a list or table containing ● Values of the variable (or a set of ranges within which the data fill) ● The corresponding frequencies with which each value occurs (or frequencies with which data fall within each range) ● Summarize data to make it useful for interpretation Example 1: your manager asks you to collect and analyze the daily sales for YNOX fertilizer in the KW area during the period Sept 7, 2018 - Sept 6, 2019 ● Continuous ratio: possible values are not countable ●

Grouping data ○ Sort from low to high (only if your doing thai by hand) ○ Find a range: 19 - 2 = 17 lbs ○ Select number of classes: 10 (usually between 5 and 20) ○ Compute class width: 2 (17/10 then round up) ○ Determine class boundaries: 2, 4,..., 20 (these are the upper bounds) ○ Count the number of values in each class

Example 2: an advertiser asks 200 customers how many days per week they use a particular mobile app ● Discrete ratio: possible values are countable ● Relative frequency = frequency/total ● Cumulative frequency = total - frequency Histograms ● The classes (or intervals or bins) are shown on the horizontal axis ● Frequencies are measured on the vertical axis ● Bars of the appropriate heights are used to represent number of observations within each class

Answer: C 5 numbers are “30 but under 40” 20 total numbers = 5/20 = 0.25

Answer: B 3 numbers above “40 but under 50” 20 tidal numbers = 20 - 3 = 17 Grouping Data into Classes ● Endpoints are determined by trial and error, subject to judgement ● The goal is to create a distribution that is neither too jagged nor too blocky ● Goal is to show pattern of variation in the data appropriately and meaningfully How Many Class Intervals? ● Many (narrow class intervals) ○ Many yield very jagged distribution with lots of gas ○ Poor indication of how frequency varies across classes

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Few (wide class intervals) ○ May compress variation too much and yield blocky distribution

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Can obscure important patterns of variation

How Many Class Intervals?

Chapter 3: Descriptive Statistics Numerical data Properties

Measures of Central Tendency: Ungrouped Data ● Provide information about the center or middle part of a group of numbers ● Provide location information ● Common measures

○ ○ ○ Mode ● ● ● ● ●

Mode Mean Median

The mode of a data set is the value that occurs most frequently Not affected by extreme values Used for either numerical or categorical data There may be NO mode if no values occurs more than once There may be several modes

Median ● Middle value in an ordered array of numbers ● Applicable for ordinal, interval, and ratio data ● Not applicable for nominal data ● Not affected by extreme values

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Mean ● ● ● ●

Recipe ○ Arrange observations in an ordered array ○ If there is an ODD number of terms, the median is the middle term ○ If there is an EVEN number of terms, the median is the average of this middle two terms ○ “Middle” Location: (n+1)/2

The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) Not applicable for nominal or ordinal data

Some Notation

Answer: D = (78 000 + 51 000)/2 = 64 500 Measures of Variability: Ungrouped Data ● Measures of variability describe the spread or the dispersion of a data set ● Common measures: ○ Range ○ Mean absolute deviation ○ Variance ○ Standard deviation ○ Coefficient variation Range ● Difference between largest and smallest ● Range = x max - x min ● ●

Ignores all other data points Simple to compute

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Sensitive to outliers

Mean Absolute Deviation ● Average of the absolute deviations from the mean (no negative numbers)

Population Variance ● Average of the squares of deviations from the mean

Population Variance Interpretation ● Variance is given in square units ● Suppose our observations are in dollars ● The variance is in squared dollars ● Difficult to interpret ● Use the square root of the variance to have intuitive interpretation ● This is called standard deviation Population Standard Deviation ● Square root of the population variance ● Sme units as observation

Sample Variance and Sample Standard Deviation ● Same idea as population variance, but related to a sample ● The sample mean x is calculated from the sample dat ● When computing SD of the same sample, we lose one “degree of freedom”

Answer: B

Comparing Standard Deviations

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The standard deviation can be used to compare the variability od several distributions If the histogram is bell shaped, we know that ○ Approx. 68% of all observations fall within 1xSD of the mean ○ Approx. 95% of all observations fall within 2xSD of the mean ○ Approx 99.7% of all observation fall within 3xSD of the mean This is called The Empirical Rule

Interpreting Standard Deviation

Z-Scores ● The z-score associated with an observation is that number’s deviation from the mean expressed in terms of standard deviations

Coefficient of Variation ● Useful for comparing variables measured in different units or with different means ● ● ●

A unit-free measure of dispersion Expressed as a percent of the mean Only appropriate for non-negative data

Chapter 4: Probability

Motivation ● Businesses want to be able to quantify the uncertainty of future events ○ The insurance industry uses probabilities in actuarial tables to determine the likelihood of certain outcomes to set specific rates and coverages ○ The gaming industry uses probability values to establish charges and payoffs ○ Teh manufacturing and aerospace industries. It is important to know the life of a mechanical part and the probability that it will malfunction at any given length of time in order to protect the firm from major breakdowns ● The study of probability helps us understand and quantify the uncertainty surrounding the future Probability: Basic Definition ● The probability of an event is a number that measures the relative likelihood that the event will occur ● The probability of event A, denoted by P(A), must lie within the interval from 0 to 1:

Experiment ● A process that produces an outcome ● More than one possible outcome ● Only one outcome per trial ● Outcome cannot be known with certainty in advance ● Trial: one repetition of the process Event ● An outcome of an experiment ● May be an elementary event: an event that can’t be further decomposed ○ Roll a die ● May be an aggregate event Sample Space ● Is a complete list of all elementary events for an experiment ● Usually denied by 5

Answer: C Methods of Assigning Probability ● Classical method: based on historical data ● Relative frequency of occurrence: based on historical data ○ Relative frequency of occurrence = Number of times an event occurred / total of trials ● Subjective probability: based on intuition, reasoning, problem intelligence ○ Useful in single trial situations such as new product introduction, IPOs, site selection, adn sporting events Classical Method ● Experiment: ○ Roll a die ○ Probabilities: each outcome has a ⅙ chance of occurring ● An experiment has n possible equally-likely outcomes Relative Frequency Method ● Based on experimentation or historical data ● Probability that a customer that is pre-approved for a mortgage will actually take out the mortgage is 95% ● Probability that a photocopier will breakdown ina given week is 30% Subjective Method ● Subjective probability comes from a person’s intuition or reasoning ● However, different individuals may assign different numeric probabilities to the same event ● Useful for unique (single trial) experiments ○ New product introduction ○ Initial public offering of common stock ○ Site selection decisions ○ Sporting events

Answer: D

Chapter 4b: More Probability Bitch Four Laws of Probability ● Law of addition ● Law of Conditional Probability ● Law of multiplication ● Bayes’ Rule

General Law of Addition

Answer: B

Motivation ● Businesses want to be able to quantify the uncertainty of future events ○ The insurance industry uses probabilities in actuarial tables to determine the likelihood of certain outcomes to set specific rates and coverages ○ The gaming industry uses probability values to establish charges and payoffs ○ Teh manufacturing and aerospace industries. It is important to know the life of a mechanical part and the probability that it will malfunction at any given length of time in order to protect the firm from major breakdowns ● The study of probability helps us understand and quantify the uncertainty surrounding the future Probability: Basic Definition ● The probability of an event is a number that measures the relative likelihood that the event will occur ● The probability of event A, denoted by P(A), must lie within the interval from 0 to 1:

Experiment ● A process that produces an outcome ● More than one possible outcome ● Only one outcome per trial

● ●

Outcome cannot be known with certainty in advance Trial: one repetition of the process

Event ● An outcome of an experiment ● May be an elementary event: an event that can’t be further decomposed ○ Roll a die ● May be an aggregate event Sample Space ● Is a complete list of all elementary events for an experiment ● Usually denied by 5

Answer: C Methods of Assigning Probability ● Classical method: based on historical data ● Relative frequency of occurrence: based on historical data ○ Relative frequency of occurrence = Number of times an event occurred / total of trials ● Subjective probability: based on intuition, reasoning, problem intelligence ○ Useful in single trial situations such as new product introduction, IPOs, site selection, adn sporting events Classical Method ● Experiment: ○ Roll a die ○ Probabilities: each outcome has a ⅙ chance of occurring ● An experiment has n possible equally-likely outcomes Relative Frequency Method ● Based on experimentation or historical data ● Probability that a customer that is pre-approved for a mortgage will actually take out the mortgage is 95% ● Probability that a photocopier will breakdown ina given week is 30% Subjective Method ● Subjective probability comes from a person’s intuition or reasoning ● However, different individuals may assign different numeric probabilities to the same event ● Useful for unique (single trial) experiments...