Introduction to Statistics With SAS

8/4/2019 Introduction to Statistics With SAS

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Section 1.1

Fundamental StatisticalConcepts


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Objectives

Explain the purpose of statistics.

Decide what tasks to complete before you

analyze your data.

Distinguish between populations andsamples.


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W

hat Is Statistics?

HEIGHT

5d4t5d10t

5d2t5d8t 5d8t

6d1t

5d5t

6d5d11t

5d


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D

escriptive Statistics

MIN AV RA = d5t MAX

I

5d5d2t 5dt

5d5t 5d8t5d8t 5d10t

5d11t 6d6d1t

5' 7.3''


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Inferential Statistics

MIN AV RA =5d5t MAX

5d5d2t 5dt5d5t 5d8t

5d8t 5d10t 5d11t 6d 6d1t

5' 7.3''


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D

efining the Problem

Before you begin any analysis, you shouldcomplete certain tasks.

1. Outline the purpose of the study.

2. Document the study questions.

3. Define the population of interest.

4. Determine the need for sampling.

5. Define the data collection protocol.


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Cereal Example

Rise

nShine

15 ounces


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Defining the Problem

The purpose of the study is to determinewhether Rise n Shine cereal boxes contain 15

ounces of cereal.The study question is whether the averageamount of cereal in Rise n Shine boxes is equalto 15 ounces.


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Population

Ri

i

Ri

i

Ri

i

Ri

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Ri

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Ri

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Ri

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Ri

i

Ri

i

Ri

i

Ri

i

Ri

i

Ri

i

Ri

i

Ri

i

Ri

i


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Sample

Risen

Shine

Risen

Shine

Risen

ShineRise

nSh in e R ise

n

Shine

Risen

ShineRise

nShine RisenShine

Risen

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nShine

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Rise

nShine Rise

nShine

Risen

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Risen

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Risen

Shine

Risen

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RisenShine

Risen

Shine


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Simple Random Sampling

Ri

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Ri

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Ri

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Ri

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Ri

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Ri

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Ri

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Ri

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Ri

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Ri

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Ri

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Ri

i...


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Convenience Sampling

Rise

nShine

Risen

Shine

Risen

Shine

Rise

nShine

Rise

nShine

Rise

nShine

Rise

nShine

Rise

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nShine

Rise

nShine...


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Parameters and Statistics

Statistics are used to approximate populationparameters.

Population

Parameters

Sample

Statistics

Mean Q x

Variance

W s

2

StandardDeviati n

W s


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Levels of Measurement

The two levels of measurement of data used inthis course are

continuous

discrete.


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Describing Your Data

The goals when you are describing data are to

screen for unusual data values

inspect the spread and shape of continuousvariables

characterize the central tendency

draw preliminary conclusions about yourdata.


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Process ofData Analysis

Popul tion

Sampl

Stati ti

Random

Sampl

D rib

Make

Inferences


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Section 1.2

ExaminingDistributions


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Objectives

Examine distributions of data.

Explain and interpret measures of location,

dispersion, and shape.

Use the MEANS and UNIVARIATE proceduresto produce summary statistics.

Use the UNIVARIATE procedure to generatestem-and-leaf, box-and-whisker, normalprobability plots and histograms.


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Cereal Data Set

Rise

nShine WEIGHT ID

.

.

.

.

.

.

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.

.

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.

.

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.

.

NUMBER


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Skewed Distributions

WEIGHT

R

EQ

UE

NC

Y


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Measures of Central Tendency

The mean is the balancing point of your data.

15.00

15.02 14.98 15.01 14.99


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Percentiles

WEI

FRE

QUE

N

CY

0

0Percentile

th

40% 60%


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Measures of Shape

FR

EQ

UE

NCY

SkewedtoRight

FR

EQ

UE

NCY

Symmetric

FR

EQ

UE

NCY

WEI WEI WEI

Skewedto Left


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Measures of Shape

Light-tailed

Normal

Heavy-tailed


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The MEANS Procedure

PROC MEANS DATA=SAS-data- et;VARvariable ;

RUN;


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The UNIVARIATE Procedure

PROC UNIVARIATE DATA=SAS-data- et;VARvariable ;

IDvariable;HISTOGRAM variable /;

PROBPLOT variable /;RUN;


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Descriptive Statistics

This demonstration illustrates using theMEANS and UNIVARIATE procedures tocalculate descriptive statistics forcontinuous variables.


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GraphicalDisplays ofDistributions

PROC UNIVARIATE produces three kinds of plotsfor examining the distribution of your data values:

stem-and-leaf plots

box-and-whisker plots

normal probability plots.

PROC UNIVARIATE can also generate histogramsand graphically enhanced normal probability plots.


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Stem-and-Leaf Plots

9 013388 00123477897 00134556677996 035685 84

3 92 01 4

Multiply Stem.Leafby 10**1


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Box-and-Whisker Plots

The mean is denoted by +.

100||

0||

80||

0||60|

|50|

|40|

|

0||20|

|10|

-

-

-

--

-

-

--

-

0

**

+

more than IQ n its from box

more than 1.5 IQ nits from box

75th percentile50th percentile median

25th percentile

max point 1.5 IQ nits from box

min point 1.5 IQ nits from box


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ExaminingDistributions

This demonstration illustrates using PROCUNIVARIATE to generate stem-and-leaf,box-and-whisker, normal probability plotsand histograms.


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Section 1.3

Confidence Intervals

for the Mean


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Objectives

Explain and interpret the confidence intervalsfor the mean.

Explain the central limit theorem.

Calculate confidence intervals usingthe MEANS procedure.


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Point Estimates

estimates

estimates


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Variability among Samples

mean o 15 02

mean o 15 0

.

.

.

.

.

.


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Standard Error of the Mean

A statistic that measures the variability of yourestimate is the standard error of the mean.

It differs from the sample standard deviationbecause

the sample standard deviation deals with thevariability of your data

the standard error of the mean deals with thevariability of your sample mean.


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( | | )

95% Confidence

| ( | )

5% Confidence


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Assumptions about


The types of confidence intervals in this coursemake the assumption that the sample means arenormally distributed.


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Distribution of Sample Means

We ight Mean ofWeight


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NormalDistribution

Useful ro abilitiesforNormalDistributions

68%95%99%

QW QW QW QWQW QQW


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x

Distribution of the SampleMeans

95%


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This demonstration illustrates calculatingconfidence intervals using PROC MEANS.


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Objectives

Define some common terminology relatedto hypothesis testing.

Perform hypothesis testing using theUNIVARIATE procedure.

Compare the means of paired groups usingthe TTEST procedure.


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Judicial Analogy

Hypothesis

CollectEvidence ecisionRule

Significance Level


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Coin Analogy

Hypothesis

CollectEvidence ecisionRule

Significance Level


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Types of Errors

You used a decision rule to make a decision, butwas the decision correct?

ACTUAL

DECISION Fair Coin Not Fair Coin

Fair Coin correct Type II error

Not Fair Coin Type I error correct


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Modified Coin Experiment

Which coins are fair?

6 H e a d s3 7 Ta i l s

4 0 H e a d s6 0 Ta i ls

5 5 H e a d s4 5 Ta i l s

1 5 H e a d s8 5 Ta i ls

p -v a lu e = .2 7

p -v a lu e < .01 p -va lu e < .01

p -va lu e = .04


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Statistical Hypothesis Test

H : equality

H :d ifference

o

1

DecisionRule

Significance Level

Collect Data

p-valuep-value

SetHypothesis

set

Risen

Shine15 oz.


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Comparing E and the p-Value

In general, you

reject the null hypothesis if p < E

fail to reject the null hypothesis if p u E.


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Performing a Test of Hypothesis

To test the null hypothesis H0: Q = Q0, SASsoftware calculates the tstatistic

xs

xt

)(0

Q!


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Two-Sided Test of Hypothesis

-3 -2 -1 0 1 2 3T


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One-Sided Test of Hypothesis

In many situations, you are only interestedin one direction. Perhaps you only want evidencethat the mean is significantly lower than fifteen.

For example, instead of testing

H0: Q = 15 versus H1: Q { 15

you test

H0: Q u 15 versus H1: Q < 15


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One-Sided Test of Hypothesis

-3 -2 -1 0 1 2 3

T


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Hypothesis Testing

This demonstration illustrates using PROCUNIVARIATE to perform hypothesis testing.


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Paired Samples

Sa lesSa les

BEFORE AFTER

ADVER T ISIN


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The TTEST Procedure

PROC TTEST DATA=SAS-data-set;CLASS variable;VARvariables;

PAIRED variable*variable;RUN;


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Paired t-Test

This demonstration illustrates using PROCTTEST to conduct a paired sample t-test.


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Section 1.5

Two-Sample t-Tests


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Objectives

Recognize and validate the assumptions ofa two-sample t-test.

Analyze two populations with the TTESTprocedure.


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Cereal Example

RisenShine

Morn

in


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Assumptions

independent observations normally distributed data for each group equal variances for each group.

Comparin T o opulations

Mornin Rise n Shine

1

2


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Test Statistics and p-Values

FTest for equal variances: H0: W12 = W2

2

Variance Test:

F = 1.51 DF = (3,3) Prob > F = 0.7446

t-Tests for equal means: H0: Q1 = Q2

Unequal Variance t-test:

T = 7.4017 DF = 5.8 Prob > |T| = 0.0004

Equal Variance t-test:

T = 7.4017 DF = 6.0 Prob > |T| = 0.0003


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Test Statistics and p-Values

FTest for equal variances: H0: W12 = W2

2

Variance Test:

F = 15.28 DF = (9,4) Prob > F = 0.0185

t-Tests for equal means: H0: Q1 = Q2

Unequal Variance t-test:

T = -2.4518 DF = 11.1 Prob > |T| = 0.0320

Equal Variance t-test:

T = -1.7835 DF = 13.0 Prob > |T| = 0.0979


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Section 1.6

OutputDelivery System


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Objectives

Introduce the Output Delivery System (ODS).

Examine some simple statements in ODS.

Use ODS to capture some specificUNIVARIATE procedure output.

Use ODS to generate a report in the HTMLformat.

Use ODS to generate data sets with specificPROC UNIVARIATE output.


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SASprocedure

computesresults

Outputobject

created inODS

ODSconverts data

componentinto SASdata set


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ODS Statements

TRACEprovides information about the output objectsuch as the name and path.

LISTINGopens, manages, or closes the Listingdestination.

OUTPUTcreates SAS data set from an output object.


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This demonstration illustrates the OutputDelivery System by introducing somesimple concepts and building on thatknowledge.


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Section 1.7

Exercises


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Section 1.8

Chapter Summary


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Section 1.9

Solutions to Exercises

Documents

Introduction to Statistics With SAS