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

Poisson Distribution

Probability distribution for the number of arrivals during each time period

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)

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.

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)

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

Data

Observed values of variables

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

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

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

Statistics

helps to make inferences and reach decisions in face of uncertainty or incomplete information.

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

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.

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

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.

Standard deviation of a discrete random variable

sqrt. [∑(xi-E(x))²×P(xi)]

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

Variance of a discrete random variable

σ²=∑(xi-E(x))²×P(xi)

Covariance formula

∑(xi-E(x)) x (Yi-E(y)) x P(XiYi)