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Statistics

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A way to get information from data

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Business Analytics

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The scientific process of transforming data into insights for making better business decisions

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The 3 major types of measurement scales

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Nominal, Ordinal, Interval

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Nominal Data

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Qualitative or categorical and labels are used to denote the classes/categories (ex. Students are classified by school such as Business, Humanities, Education, and so on)

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Ordinal Data

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Same as nominal data but there is ordering or ranking (ex. Credit ratings such as Excellent, Good, Fair)

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Interval Data

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Quantitative or numerical in nature (ex. SAT scores, exam grades, income, height, weight, etc)

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All arithmetic operations are possible for interval data but not for nominal and ordinal data types

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True

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Cross-sectional data

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Data is collected at the same or approximately the same point in time

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Time Series Data

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Collected over several time periods (ex. price of gas over a period of years)

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The two types of Statistics

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Descriptive, Inferential

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Descriptive Statistics

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Refers to the summary of important aspects of a data set

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Inferential Statistics

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Goes beyond the data at our disposal (More formally refers about large data set "population" based on smaller set of "sample" data

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Using a survey of a random sample of 5000 California residents, an economist said over 55% have a positive view of the economy

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Inferential Statistics

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Say, from a survey of a random sample of 5000 students, 80% of those sampled are very excited about stats

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Descriptive Statistics

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Population

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A set of items (experimental units) under study

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Parameter (Variable)

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A descriptive measure of the population that is of interest e.g. the mean (Unknown - Use greek letter)

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Statistic (singular term)

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A descriptive measure that is calculated from the sample e.g. the sample mean

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The purpose of Inferential Stats

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To make inferences about a parameter of a population based on information obtained from the statistic or sample (with a certain degree of confidence)

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The goal of data collection

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To obtain a "representative sample" that exhibits the characteristics of the entire population, most common approach is taking random samples

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Sources of Statistical Data

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Extract from a public source, perform a designed experiment, take a survey, perform an observational study

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Non-Random Sampling Errors

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Selection Bias, Non-response Bias, Measurement Errors

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Three key features to understanding the shape of the data

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Symmetry, Skewness, Modality

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Symmetry

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Normal distribution, the curve is shaped like a bell in the middle

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Skewness, Skewed

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The bell will be on the left (negative skew) or to the right (positive skew)

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Modality

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The curve will tell you if there are more than one groups in your data, camel humped curve

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Mean

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The simple average

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Median

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The middle observation after the data has been orderd

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Mode

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The observation that occurs most often

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Range=

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Largest value-smallest value

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Interquartile Range=

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3rd quartile-1st quatile =Q3-Q1

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Z Score

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Used to measure the location of a particular value in the data relative to the mean, the bigger the score, the farther from the mean/average

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Empirical Rule

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For a normal distribution, nearly all of the data will fall withing 3 standard deviations of the mean

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Outliers

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Values that fall outside of the normal range, either unusually large or unusually small

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Chebyshev's Theorem

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For z>1, at least (1-1/z^2)100% of the data values must be withing z standard deviations of the mean

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Random Variable

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A numerical description of the outcome of an event (ex. The amount a company pays out on an individual policy, based on the outcome of a random event)

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Probability

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The likelihood that an event will occur and give values within a particular range

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Probability Distribution

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The collection of all possible values of the random variable X and the associated probabilities P(X=x), always between 0 and 1. The sum of all P(X=x) is always 1

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Central Limit Theorem

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Even if X does not have a normal distribution, X will be approximately normal if n is large (n=30 is usually large enough)

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p bar is the point estimator or sample statistic for the population parameter; p

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pbar=x/n

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Point Estimator

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The value (obtained from a sample) which is considered a best guess or estimate of a population parameter

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Parameter

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We have a single population "target of interest". From this population, we identify a parameter. This parameter is a fixed numerical value. The problem is that we do not know its value, but wish to know it. Interval estimation will help us estimate it.

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Sampling distribution of xbar

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Tells us (in probability of terms) how close a point estimator is to the parameter

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Margin of Error (ME or MOE)

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ME or MOE is a quantification of how close a point estimator is to the parameter.

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Is it possible that an interval estimate may not capture the parameter?

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Yes. This is what we call uncertainty.

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Is there a way of controlling the uncertainty that the interval captures the parameter?

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Yes. Attach a level of desired certainty to the interval estimate.

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Confidence Level or (1-alpha) (ex. we are 95% confident)

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Quantifies how often a confidence interval captures the population parameter

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Margin of Error

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Critical Value x Standard Error

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Confidence Level and ME work in opposite directions

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If we want to be more confident (a higher confidence level), we have to accept a higher ME. A higher ME means that our interval estimate will be less precise.

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A large n

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Has clear benefits as ME is lower but may cost us time and money

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A small n

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Leads to a higher ME, so no real benefit but costs less

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The compromise

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The compromise will be to specify some desired ME or MOE and look for n that can achieve our specified goal

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Null Hypothesis

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The hypothesis that can possibly be disproved using sample information or evidence

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Type 1 Error

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Occurs when rejecting the null hypothesis when it is actually true, or claiming the alternate when it is not true

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Type 2 Error

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Occurs when we fail to reject the null hypothesis when it is actually false

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When using p-value approach

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Reject the null if p-value is less than alpha, and do not reject null if p-value is more than alpha

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Critical Value Approach

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If the Critical Value is a positive, reject null if test statistic is greater than CV. IF CV is negative, reject null if test statistic is less than CV.

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Matched Sample Design

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Each sampled item provides a pair of data values. This design often leads to a smaller sampling error than independent-sample design because it is a variation between sampled items is eliminated as a source of sampling error.

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Experiment

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A study in which the experimenter manipulates attributes of what is being studied and observes the consequences.

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Factors

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These are the attributes that are manipulated by being set to particular levels and then assigned to individuals. An experimenter identifies at least one factor to manipulate. these levels are often called treatments.

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Observed Response

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A quantitative measurement in ANOVA

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ANOVA

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Analysis of Variance, can be used to test for the equality of three or more population means

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Data obtained from observational or experimental studies can be used for the analysis

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True

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If Ho is rejected, we cannot conclude that all population means are different

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True

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Rejecting Ho means that at least two population means have different values

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True

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For each population, the response (dependent) variable is normally distributed

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True

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ANOVA can be viewed as the process of partitioning the total sum of squares and the degrees of freedom into their corresponding sources: treatments and error

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True

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Randomized Block Design

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Experimental units are the objects of interest in the experiment

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Completely randomized design

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An experimental design in which the treatments are randomly assigned to the experimental units

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Factorial Experiment

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Used because the conditions include all possible combinations of the factors.

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Managerial decisions often are based on the relationship between two or more variables

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True

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Regression Analysis

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Can be used to develop an equation showing how the variables are related

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Dependent Variable

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Variable being predicted and denoted by "y"

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Independent Variable

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Variable being used to predict value of the dependent variable and denoted by "x"

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Simple Linear Regression

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Involves one independent variable and one dependent variable, approximated by a straight line

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Regression Model

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Equation that describes how "y" is related to "x"

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Simple linear regression model

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y = Bo + B1x + E

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Bo and B1

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Parameters of the model

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Bo

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The "y" intercept of the regression line

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B1

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The slope of the regression line

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Estimated Simple Linear Regression Equation

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yhat = Bo +B1x

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