Economic Statistical Analysis Final Review
Who and when might someone use Statistics examples:
Businessperson: to test the product design or package that maximizes sales. Sociologist: to analyze the result of a drug rehabilitation program. Political Scientist: to forecast voting patterns. Physician: to test the effectiveness of a new drug How does Netflix (or Spotify) know what kind of movies (music) do you like?
Counting Rule Combinations (The number of ways of selecting X objects from n objects, irrespective of order) Ex: You have five books and are going to select three are to read. How many different combinations are there, ignoring the order in which they are selected?
C=(n!)/X!(n-X)! Ex: (5!)/3!(5-3)! 120/(6(2!)) 120/12 10
Discrete Random Variables
Can only assume a countable number of values
Variables
Characteristics of items or individuals
Two events are independent if
P(A|B)=P(A) when the probability of one event is not affected by the fact that the other event has occurred.
Counting Rule: Permutations (The number of ways of arranging X objects selected from n objects in order is) EX: You have five books and are going to put three on a bookshelf. How many different ways can the books be ordered on the bookshelf?
P=(n!)/(n-X)! (n=total) EX: (5!)/(5-3)! 120/2 60
General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
Binomial Probability distribution requirements
1. The procedure has a fixed number of trials 2. The trials must be independent 3. Each trial must have all outcomes classified into two categories 4. The probability of a success remains the same in all trials 5. Observations are independent
inferential statistics
A branch of mathematics that involves drawing conclusions about a population based on sample data drawn from it.
Applications for the binomial distribution
A manufacturing plant labels its items as either defective or acceptable A firm bidding for a contract will either get the contact or not marketing firm receives survey responses of yes or no New job applicants either accept job offer or reject it
Binomial Probability Distribution
A probability distribution showing the probability of x successes in n trials of a binomial experiment.
Which method of data measurement allows for rank order (1st, 2nd, 3rd, etc.) by which data can be sorted but not for a relative degree of difference between them?
Ordinal
Multiplication rule for two independent events A and B
P(A and B)=P(A)P(B)
Which method of data measurement has meaningful distances between measurements defined, but an arbitrary zero value?
Interval
Counting rule: K1 trial, K2 trial... Ex: You want to go to a park (3), Eat at a resturant (4) and see a movie (6 movies avail) how many combos are there?
K1(k2)(K3).... EX: 3x4x6=72
Counting Rule: K is mutually exclusive and collectively exhaustive events EX: you roll a die 3x how many outcomes are there
K^n EX: 6^3=216 possible outcomes
What does the shape of the poisson distribution depend on?
Lamda
Investment Objective
Maximize return (mean) while minimizing risk (standard deviation).
Which method of data measurement differentiates between items or subjects based only on qualitative classifications they belong to?
Nominal
Scales of Measurements
Nominal Ordinal Interval Ratio The scale determines the amount of information contained in the data
Which method of data measurement has both a meaningful zero value and a defined distance between different measurements?
Ratio
Calculate Expected Value (Mean) of Discrete Random Variables
Sum of X(P(Xi) X1(P(X1))+X2(P(X2))+....Xn(P(Xn))
Contingency Tables
Table used to classify sample observations according to two or more identifiable characteristics
statistics
The collection, presentation, analysis and utilization of numerical data
Data Set
The data collected in a particular study
Primary Sources
The data collector is the one using the data for analysis Example:Data from a political survey, Data collected from an experiment, Observed data
Mutually exclusive events
The occurrence of one event means that none of the other events can occur at the same time
Secondary Sources
The person performing data analysis is not the data collector Example: Analyzing census data, Examining data from print journals or data published on the internet.
Binomial Distribution
The probability distribution of X with parameters n and p
observation
The set of measurements collected for a particular element
Sampling
To use a sample as a guide to an entire population, it is important that it truly represent the overall population.
Certain Event
an event that is sure to occur (probability = 1) The closer the probability is to one, the more sure we are the event will happen
probability distribution for a discrete random variable
a mutually exclusive listing of all possible numerical outcomes for that variable and a probability of occurrence associated with each outcome.
negative covariance indicates....
a negative relationship
positive covariance indicates...
a positive relationship
variable
a quantity that may assume any one of a set of values
random sample
a sample randomly taken from an investigated population
Ratio Data
a type of numerical data in which the difference between numbers is significant, but there is a fixed non-arbitrary zero point associated with the data Example: Weight and height
Critical thinking
a way of deciding whether a claim is always true, sometimes true, partly true, or false
Impossible Event
an event that has no chance of occurring (probability = 0) The closer the probability is to zero, the more improbable it is the event will happen
Subjective probability
based on a combination of an individual's past experience, personal opinion, and analysis of a particular situation
A priori Approach to probability
based on prior knowledge of the process (Number of favorable outcome)/(Total number of possible outcome)
empirical verifiable
by means of scientific experimentation
conditional probability
the probability of a particular event to occur, given that another event has occurred: P(A|B)=P(A and B)/P(B)
ordinal data
data exists in categories that are ordered but differences cannot be determined or they are meaningless. Example: 1st, 2nd, 3rd/Distinction, Merit, Pass or Fail.
primary data
data that has been compiled for a specific purpose, and has not been collated or merged with other
Simple Probability
the probability of a simple event Ex: Probability of selecting a kind
independent variable
in an equation, any variable whose value is not dependent on any other in the equation
Joint probability
the probability that measures the likelihood that two or more events will happen concurrently
Counting rule: The number of ways that n items can be arranged in order is EX: How many ways can you arrange 5 books on a self
n! 5x4x3x2x1=120
Binomial Distribution Mean
n(pi)
Binomial Distribution Variance and Standard Deviation
n(pi)[(1-Pi)] SD: sqrt
qualitative data
nominal or ordinal, categorical data Information describing color, odor, shape, or some other physical characteristic can be either numeric or nonnumeric
descriptive statistics
numerical data used to measure and describe characteristics of groups. Includes measures of central tendency and measures of variation. Concerned with summarizing and describing a body of data Describing what was observed in the sample numerically or graphically
Standard deviation of a discrete random variable
sqrt. [∑(xi-E(x))²×P(xi)]
Variance of a discrete random variable
σ²=∑(xi-E(x))²×P(xi)
Complement Rule
Used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.
variable
a characteristic of an item or individual
area of opportunity
a continuous unit or interval of time, volume, or such area in which more than one occurrence of an event can occur. EX: The number of scratches in a car's paint The number of mosquito bites on a person The number of computer crashes in a day
population
a group of units (persons, objects, or other items) enumerated in a census or from which a sample is drawn
Statistics
helps to make inferences and reach decisions in face of uncertainty or incomplete information.
Probability
is the study of events and outcomes involving an element of uncertainty. Ex: Investing in stock markets, Flipping a coin
What does it mean if something has a high standard deviation?
it is subject to much more variability and the probability of loss is higher
covariance
measures the strength of the linear relationship between two discrete random variables X and Y.
Joint Probability
probability of an occurrence of two or more events Ex: Probability of selecting a king and spade
Discrete random variables
produce outcomes that come from a counting process (e.g. number of classes you are taking).
Continuous random variables
produce outcomes that come from a measurement (e.g. your annual salary, or your weight).
random variable
represents a possible numerical value from an uncertain event.
Statistical literacy
the ability to read and interpret statistics and to think critically about arguments that use statistics as evidence necessary to understand what makes a poll trustworthy and to properly weigh the value of poll results and conclusions.
critical thinking
the application of logical principles, rigorous standards of evidence, and careful reasoning to the analysis and discussion of claims, beliefs, and issues
Sample Space
the collection of all possible events
elements
the entities on which data are collected
Sample
the portion of the population selected for analysis
Inferential Statistics
the process of reaching generalizations about the whole (called population) by examining a portion (called the sample) n order for this to be valid, the sample must be representative of the population and the probability of error also must be specified. Tests of significance used to determine the probability that the results were found by chance.
sampling
the process or technique of obtaining a representative sample
dependent variable
the variable whose value depends on one or more variables in the equation
What does it mean if covariance between X and Y is positive
there is a positive relationship between the X and Y, meaning that they will likely rise and fall together.
Nominal Data
used to "name" or label values such as gender and hair color, they have no meaningful rank order among values. EX: Gender, eye, hair color
Bayes' Theorem
used to revise previously calculated probabilities based on new information. Developed by Thomas Bay 18th Century
When do you use the Poisson distribution
when you are interested in the number of times an event occurs in a given area of opportunity
Covariance formula
∑(xi-E(x)) x (Yi-E(y)) x P(XiYi)
Multiplication rule for two events A and B
P(A and B)=P(A|B)P(B)
General Addition rule for mutually exclusive events
P(A or B) = P(A) + P(B)
Poisson Distribution
Probability distribution for the number of arrivals during each time period
experiment
A test under controlled conditions made to either demonstrate a known truth, examine the validity of a hypothesis, or determine the efficacy of something previously untried.
Probability
A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur. Frequently expressed in decimal such as 0.70, 0.27.
Complement of an event A (denoted A')
All events that are not part of event A e.g., All cards that are not diamonds
Simple event
An event described by a single characteristic e.g., A red card from a deck of cards
Joint event
An event described by two or more characteristics e.g., An ace that is also red from a deck of cards or the electricity goes out AND the generator fails
Collectively exhaustive events
At least one of the events must occur when an experiment is conducted The set of events covers the entire sample space
Empirical Approach to probability
Based on frequency of the event in the past (Number of times the even occurred in the past)/(Total number of possible outcomes)
Population
Consists of all the members of a group about which you want to draw a conclusion
rule of combinations EX: How many possible 3 scoop combinations could you create at an ice cream parlor if you have 31 flavors to select from
Counting technique use when determining The number of combinations of selecting X objects out of n objects is n!/[X!(n-X)!] 31!/3!(31-3)! 31x5x29 4495
Quantitative Data
Data measuring how much, and be discrete or continuous, always numerical, ratio and interval data
continuous data
Data that can take on any value. There is no space between data values for a given domain. Example: Height/Weight
discrete data
Data with space between possible data values. Example: How many apples do I have
Interval Data
Differences between values can be found, but there is no absolute 0. Example: Temp. and Time
Expected Value of the sum of two random variable formula
E(X+Y)=E(X)+E(Y)
Data
Observed values of variables
