Chapter 1

Section 1.1

Objectives
• Explain the purpose of statistics.
• Decide what tasks to complete before you analyze your data.
• Distinguish between populations and samples.

What Is Statistics?
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Descriptive Statistics
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 MAX

Inferential Statistics
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Defining the Problem
Before you begin any analysis, you should complete 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.

Cereal Example
Rise n
Shine
15 ounces

Defining the Problem
The purpose of the study is to determine whether Rise n Shine cereal boxes contain 15 ounces of cereal.
The study question is whether the average amount of cereal in Rise n Shine boxes is equal to 15 ounces.

Population
Rise n
Shine
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Rise

Sample
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Shine
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Shine Rise n
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Shine Rise n
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Simple Random Sampling
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Rise ...
n
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R ise n
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Convenience Sampling
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...
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R ise n
Shine
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Shine

Parameters and Statistics
Statistics are used to approximate population parameters.
Mean
Variance
Standard
Deviation
Population
Parameters

 2

Sample
Statistics x s 2 s

Levels of Measurement
The two levels of measurement of data used in this course are
• continuous
• discrete.

Describing Your Data
The goals when you are describing data are to
• screen for unusual data values
• inspect the spread and shape of continuous variables
• characterize the central tendency
• draw preliminary conclusions about your data.

Process of Data Analysis
Population
Random
Sample
Describe
Sample
Statistics
Make
Inferences

Section 1.2

Objectives
• Examine distributions of data.
• Explain and interpret measures of location, dispersion, and shape.
• Use the MEANS and UNIVARIATE procedures to produce summary statistics.
• Use the UNIVARIATE procedure to generate stem-and-leaf, box-and-whisker, normal probability plots and histograms.

Cereal Data Set
Rise n
Shine
.
.
.
.
.
.
WEIGHT
.
.
.
.
.
.
ID NUMBER
.
.
.
.
.
.

Distributions
When you examine the distribution of values for the variable WEIGHT, you can find out
• the range of possible data values
• the frequency of data values
• whether the data values accumulate in the middle of the distribution or at one end.

Symmetric Distributions
WEIGHT

Skewed Distributions
WEIGHT

Normal Distribution
Examples of Normal Distributions std 1.5 std 1.0 std 0.5

Measures of Central Tendency
The mean is the balancing point of your data.
15.02
14.98
15.01
14.99
15.00

Percentiles
40 th
Percentile
0
40% 60%
WEIGHT

Measures of Dispersion
15.00
WEIGHT
15.00
WEIGHT

Measures of Shape
Skewed to Left Symmetric
Skewed to Right
WEIGHT WEIGHT WEIGHT

Measures of Shape
Light-tailed
Normal
Heavy-tailed

The MEANS Procedure
PROC MEANS DATA=SAS-data-set <options>;
VAR variables;
RUN;

The UNIVARIATE Procedure
PROC UNIVARIATE DATA=SAS-data-set<options>;
VAR variables;
ID variable;
HISTOGRAM variables / <options>;
PROBPLOT variables / <options>;
RUN;

Descriptive Statistics
This demonstration illustrates using the
MEANS and UNIVARIATE procedures to calculate descriptive statistics for continuous variables.

Graphical Displays of Distributions
PROC UNIVARIATE produces three kinds of plots for examining the distribution of your data values:
• stem-and-leaf plots
• box-and-whisker plots
• normal probability plots.
PROC UNIVARIATE can also generate histograms and graphically enhanced normal probability plots.

Stem-and-Leaf Plots
9 01338
8 0012347789
7 0013455667799
6 03568
5 8
4
3 9
2 0
1 4
Multiply Stem.Leaf by 10**1

Box-and-Whisker Plots
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0
*
* max point 1.5 IQ units from box
75th percentile
50th percentile median
25th percentile min point 1.5 IQ units from box more than 1.5 IQ units from box more than 3 IQ units from box
The mean is denoted by +.

Normal Probability Plots
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Examining Distributions
This demonstration illustrates using PROC
UNIVARIATE to generate stem-and-leaf, box-and-whisker, normal probability plots and histograms.

Section 1.3

Objectives
• Explain and interpret the confidence intervals for the mean.
• Explain the central limit theorem.
• Calculate confidence intervals using the MEANS procedure.

Point Estimates estimates estimates

Variability among Samples
.
.
mean of 15.02
mean of 15.03
.
.

Standard Error of the Mean
A statistic that measures the variability of your estimate is the standard error of the mean.
It differs from the sample standard deviation because
• the sample standard deviation deals with the variability of your data
• the standard error of the mean deals with the variability of your sample mean.

Confidence Intervals
95% Confidence
( | | )
5% Confidence
| | )

Assumptions about
Confidence Intervals
The types of confidence intervals in this course make the assumption that the sample means are normally distributed.

Distribution of Sample Means
Weight Mean of Weight

Normal Distribution
Useful Probabilities for Normal Distributions
68%
95%
99%
      

Confidence Intervals
Distribution of the Sample Means
95% x

Central Limit Theorem
To satisfy the assumption of normality, you can either
• verify that the population distribution is approximately normal, or
• apply the central limit theorem.
The central limit theorem states that the distribution of sample means is approximately normal provided that the sample size is large enough.

Central Limit Theorem

Confidence Intervals
This demonstration illustrates calculating confidence intervals using PROC MEANS.

Section 1.4

Objectives
• Define some common terminology related to hypothesis testing.
• Perform hypothesis testing using the
UNIVARIATE procedure.
• Compare the means of paired groups using the TTEST procedure.

Judicial Analogy
Hypothesis Significance Level
Collect Evidence Decision Rule

Coin Example
H
T
H
T
H

Coin Analogy
Hypothesis Significance Level
Collect Evidence Decision Rule

Types of Errors
You used a decision rule to make a decision, but was the decision correct?
ACTUAL
DECISION
Fair Coin
Fair Coin correct
Not Fair Coin Type I error
Not Fair Coin
Type II error correct

Modified Coin Experiment
Which coins are fair?
55 Heads
45 Tails
p-value = .27
63 Heads
37 Tails
p-value < .01
40 Heads
60 Tails
p-value = .04
15 Heads
85 Tails
p-value < .01

Statistical Hypothesis Test
Set Hypothesis
Rise n
Shine
15 oz.
Collect Data set
Significance Level
p-value
p-value
Decision Rule

Comparing  and the p -Value
In general, you
• reject the null hypothesis if p < 
• fail to reject the null hypothesis if p   .

Performing a Test of Hypothesis
To test the null hypothesis H
0
:  software calculates the t statistic
= 
0
, SAS t

( x
 
0
) s x

Two-Sided Test of Hypothesis
The test of hypothesis is two-sided if the null is rejected when the actual value of interest is either less than or greater than the hypothesized value.
H
0
:   15.00
H
1
:   15.00

Two-Sided Test of Hypothesis
-3 -2 -1 0 1 2 3
T

One-Sided Test of Hypothesis
In many situations, you are only interested in one direction. Perhaps you only want evidence that the mean is significantly lower than fifteen.
For example, instead of testing
H
0
:  = 15 versus H
1
:   15 you test
H
0
:   15 versus H
1
:  < 15

One-Sided Test of Hypothesis
-3 -2 -1 0 1 2 3
T

Hypothesis Testing
This demonstration illustrates using PROC
UNIVARIATE to perform hypothesis testing.

Paired Samples
BEFORE
ADVERTISING
AFTER
Sales
Sales

The TTEST Procedure
PROC TTEST DATA=SAS-data-set;
CLASS variable;
VAR variables;
PAIRED variable*variable;
RUN;

Paired t -Test
This demonstration illustrates using PROC
TTEST to conduct a paired sample t-test.

Section 1.5
t

Objectives
• Recognize and validate the assumptions of a two-sample t-test.
• Analyze two populations with the TTEST procedure.

Cereal Example
Rise n
Shine
M o rn in g

Assumptions
Comparing Two Populations
2
1
• independent observations
Rise n Shine
• normally distributed data for each group
• equal variances for each group.

F Test for Equality of Variances
H :
0
2
1
=
2
2
H :
1
2
1
=
2
2
F=
2 2 max(s , s )
1 2
2 2 min(s , s )

Test Statistics and p -Values
F Test for equal variances: H
0
: 
1
2 = 
2
2
Variance Test:
F’ = 1.51
DF = (3,3) Prob > F’ = 0.7446
t-Tests for equal means: H
0
: 
1
= 
2
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

Test Statistics and p -Values
F Test for equal variances: H
0
: 
1
2 = 
2
2
Variance Test:
F’ = 15.28
DF = (9,4) Prob > F’ = 0.0185
t-Tests for equal means: H
0
: 
1
= 
2
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

Testing for Equality of Means
This demonstration illustrates using PROC
TTEST to test for the equality of means for two groups.

Section 1.6

Objectives
• Introduce the Output Delivery System (ODS).
• Examine some simple statements in ODS.
• Use ODS to capture some specific
UNIVARIATE procedure output.
• Use ODS to generate a report in the HTML format.
• Use ODS to generate data sets with specific
PROC UNIVARIATE output.

Output Delivery System
SAS procedure computes results
Output object created in
ODS
ODS converts data component into SAS data set

ODS Statements
• TRACE provides information about the output object such as the name and path.
• LISTING opens, manages, or closes the Listing destination.
• OUTPUT creates SAS data set from an output object.

Output Delivery System
This demonstration illustrates the Output
Delivery System by introducing some simple concepts and building on that knowledge.

Section 1.7

Section 1.8

Section 1.9