Introductory Statistical Concepts

Disclaimer – I am not an expert SAS programmer. – Nothing that I say is confirmed or denied by Texas A&M University. 2

Why Are We Here? Deming – To Learn – To Have Fun Question: Who was Deming? 3

Poll: What type of organization do you work for? [PlaceWare Multiple Choice Poll. Use PlaceWare > Edit Slide Properties... to edit.] Business Government Education Nonprofit Other 4

Purpose of These Lectures A review of the statistical concepts used in most of the SAS Analytics Lecture Series. We will look at questions such as the following: – What is the nature of statistical analyses? – Why are population parameters so important? – What is really being tested when you see a p-value? – Why does regression handle missing data so well? – What are residual analyses? 5

Descriptive Statistics

7 The Population (Very important concepts) Variable of Interest The Distribution Parameters MeanModeRange MedianVariance Etc

Learning Outcomes You will learn – basic statistical concepts – the definition of mean, median, mode and standard deviation – the difference between populations and samples – the difference between parameters and estimates – about confidence intervals – how to test a statistical hypothesis – how to run a regression analysis 8

Parameters Characteristics of the variable of interest It is how we describe the variable of interest Parameters are unknown 9

Parameters (Characteristics) Central Tendency Mode Median Mean Measures of Variability Range Variance Standard Deviation 10 Click Here Click Here for more information on Mode Mean Median Click Here Click Here for an applet

Variability Change in the Data

12 What is an Index ? How SUNNY is SUNNY? THE UV Index Click Here

13 Air Quality Index What Does It Mean?

DOW JONES INDUSTRIAL AVERAGE INDEX 14 What does 10, really mean? What is “better” a DJIA of 10,000 Or a DJIA of 12,000?

Variability Index A Simple One Find the Largest Value Find the Smallest Value Let Range = R = Largest – Smallest 15

A More Complex Variation Index The Standard Deviation Statisticians use this index to indicate variability You will see it written as Widely available from SAS, Excel, and other statistical packages 16

Details of the More Complex Index Example – Suppose that we observe the following three numbers The mean of these number is: ( )/3 = 4 We now subtract the mean from each number and square it (1-4)*(1-4) + (4-4)*(4-4) +(7-4)*(7-4) = 18 The Standard Deviation = sqrt(18/2) = 3 17

What does this Mean? By itself, it may be confusing to some. Comparing populations, we can use it to say which population varies the most. Let us look at an applet – Click HereClick Here 18

Using Graphs to Determine Variability Box Plot Click Here 19

Distributions

Known Distribution With a known distribution, we know the following: – the shape – the mean – the variability (standard deviation) – and/or some other information 21

Classical Distributions─Normal 22

Normal─Overlay 23

Classical Distributions─Uniform 24

Survey The following are called parameters of the population: – mean, median, mode – variance, standard deviation, range, inter-quartile range (IQR) In general, are these known or unknown? – Known = yes (select using your seat indicator) – Unknown = no (select using your seat indicator) 25

MPG─Histogram 26 Compare with “true” values !

Simulated Sample In this example, we simulated taking a sample of size 1000 from one population of cars weighing 3000 pounds with a normal distribution with mean=24 and standard deviation=1. You can practice this after class. 27

Section 1.2 Populations and Samples

Objectives – Understand the relationships between populations and samples parameters and estimates. – Look at an overview of hypotheses testing. 29

Population 30 Mean, Variance, Median, Mode, Distribution, … Parameters

Example Mpg of American-made cars that weigh between 2000 and 3500 pounds and were built in the 1970s. Parameters – mean, variance, and so on In general, we do not know the parameters. 31

Purpose of Statistical Analyses – Estimate the parameters. (Make guesses.) Example: What is the population mean? – Test hypothesis about the parameters. (Ask questions.) Example: Is the population mean=30mpg? 32

Role of Samples – Taking a sample of the population enables you to make estimates of the population parameters answer the questions about the population parameters. 33

Population and Sample 34 Mean, Variance, Median, Mode, Distribution, … Parameters Sample mean Sample variance Sample S Inference: Estimates Test of hypotheses

Example: cars_american This is a sample of American-made cars that weigh between 2000 and 3500 pounds and that were built in the 1970s. We are interested in the mpg. Use summary statistics to analyze the data. 35

Results of Summary Statistics 36

Results of Histogram 37 continued...

Results of Histogram 38

Sampling Distribution Applet sampling_dist This demonstration illustrates how to estimate and plot the sampling distribution of various statistics. 39

View/Application Share: Demo: Sampling Distributions Applet [PlaceWare View/Application Share. Use PlaceWare > Edit Slide Properties... to edit.] 40

m/sampling_dist/index.h... [PlaceWare Web Page. Use PlaceWare > Edit Slide Properties... to edit.] 41

Confidence Intervals on the Population Mean Level of Comfort 50% {21.57 to 22.21} 95% {20.96 to 22.82} 99.9% {20.30 to 23.48} 42 What does this mean?

Test That the Population Mean = 30 mpg Use t-test  One Sample t-test Requirements for running this test: – Large n > 35 – Or leftovers are normal What is the p-value or sig value? 43

Testing Mean = 30 44

Conclusions of the Test Choose an alpha level, usually alpha=.05. If sig<alpha, then reject. Otherwise, fail to reject. 45

Sig and p-values When you see a sig value or p-value: – You know that some hypothesis is being tested. – You know whether or not the hypothesis is being rejected. – You probably do not know what the hypothesis really is. Ask yourself these questions: – What are the population parameters being tested? – How is what is being tested related to those parameters? 46

Requirements for Doing This Test Large n  n > 35 Or leftovers are normally distributed. Use Histogram to test for normality. 47

Populations─Which Ones are Similar? 48

Populations─Which Ones are Similar? Take samples. 49

Take Samples Use the samples to answer this question: “Which populations are similar?” Statistical translations: “Which populations are similar?” is the same as asking… Are the following the same: – distribution? – mean? – variance? 50

Background/Requirements Before we jump into the analysis, we must ask the following questions: – How many populations are there? – How many population parameters are we interested in and what are they? – What tests do we want to do, and what are the requirements for doing those? – Are we using everything we “know?” 51

Example Suppose that we are interested in the mpg of American and European cars. How many populations are there? 52 American Cars Mpg Distribution Mean Variance European Cars Mpg Distribution Mean Variance

Poll: How many populations are there? [PlaceWare Multiple Choice Poll. Use PlaceWare > Edit Slide Properties... to edit.] One - MPG Two - American and European Depends on the sample size 53

Parameters Population 1Population 2 American CarsEuropean Cars Variable of interest: mpg Distribution: Normal? Mean: Variance: 54

Analyses 1.We want to look at the distributions. 2.We want to estimate the parameters. 3.We want to answer these questions: Are the populations means the same? Are the population variances the same? 55

Example: Our Data Set car_am_eu Suppose that we are interested in the mpg of American and European cars. 56 Sample American Cars Mpg Distribution Mean Variance European Cars Mpg Distribution Mean Variance Sample

Results from the Sample 57 continued...

Results 58

Box Plots 59 American European

Histograms 60 American European

Poll: Are the populations the same? [PlaceWare Yes/No Poll. Use PlaceWare > Edit Slide Properties... to edit.] Yes No 61

Conclusion Based on Sample Numbers and Graphs Easy -- Based on the samples, the populations are different—no statistical jargon But I must have a p-value for my boss, for my paper, and so on. 62

Formal Tests The classical approach in determining whether two populations are the same is to test to see whether the two population means are equal. But first we check to see whether the two population variances are equal: 63 continued...

Formal Tests We use t-test  Two Sample. 64 Test 2 Test 1

Section 1.3 Simple Linear Regression

Objectives – Identify the following: the population parameters the appropriate model number of populations sampled the correct hypotheses what should be tested for normality what “equal variances” means. 66

MPG Example 67 Weight = 3000 Weight = 2600 Weight = 2900 Weight = 2300 Take a sample of size 1 from each population!

Data We should be in deep trouble with one sample from each population. We have eight unknown population parameters. Can you name them? But what do we “know”? 68

Survey Name the population parameters. 69

Essential Part and Leftovers We want to “model” the data as follows: MPG = Essential Part + Leftover or MPG = Mean + Leftover 70

“Know” or Assumptions First, we “know” that Second, each population mean is related to weight by the following: The population means fall on a straight line!! How many unknowns are there now? 71

Poll: How many unknowns are there? [PlaceWare Multiple Choice Poll. Use PlaceWare > Edit Slide Properties... to edit.] n 72

Graph 73

Observed, Essential Part, Leftover 74

The Official Regression Model or 75 The errors are “known” to be normal with mean 0 and variance.

Main Assumptions The means of the populations fall on a straight line. All of the variances are equal ( ). The errors are “known” to be normal with mean 0 and variance. 76

Assumptions for Simple Linear Regression Appendix A This demonstration illustrates the fundamental concepts of simple linear regression. 77

View/Application Share: Demo: Linear.doc [PlaceWare View/Application Share. Use PlaceWare > Edit Slide Properties... to edit.] 78

How Can We Estimate the Unknown Parameters? The Principle of Least Squares: or Now, choose a and b so that is as small as possible. or Minimize. 79

OUTPUT_0 80

OUTPUT 81

OUTPUT_1 82

OUTPUT_2 83

OUTPUT_3 84

OUTPUT_4 85

Missing Values Suppose that we want to estimate the mean mpg when weight=2500. Predicted (Estimated) Mean MPG = *weight Why does this work? 86

Survey Can anyone explain why this works? 87

Conclusion – Simple linear regression is very powerful. – But it is based on assumptions (what we “know”). – We need to check assumptions (residual analyses). 88