1. Data Analysis and Statistical Software I (323-21-403) Quarter: Autumn 02/03
Daniela Stan, PhD
Course homepage:
Office hours: (No appointment needed)
M, 3:00pm - 3:45pm at LOOP, CST 471
W, 3:00pm - 3:45pm at LOOP, CST 471
11/22/2018
Daniela Stan - CSC323

2. Outline Chapter 6: Introduction to Inference Confidence Intervals
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3. Population: mean µ unknown
Population: mean µ unknown standard deviation  known How do we estimate µ.?
SRS1
SRS2
SRSn
The center of the distribution of the sample averages is the population average . A measure of the spread is the standard error. 
The central theorem says:
In repeated sampling,
the sample mean x follows the normal distribution centered at the unknown population mean and having standard deviation x = /(n1/2) x1 x2 xn
11/22/2018
Daniela Stan - CSC323

4. Confidence Intervals Population: mean µ unknown
Sample mean x is an unbiased estimator of µ.
How reliable/accurate is this estimator?
What is the variability of the estimator?
The rule says: In the
normal distribution with mean µ
and standard deviation :
approximately 95% of the
observations fall within 2*  of µ
Since x has a normal distribution
the rule says:
approximately 95% of all the
samples will capture the true value µ
in the interval
(x -2 /(n1/2), x+2 /(n1/2))
x ~ N(µ, /(n)1/2)
11/22/2018
Daniela Stan - CSC323

5. Confidence Intervals (cont.)
In the language of statistical inference, we say that we are 95% confident that the unknown population mean lies in the interval
(x -2/(n1/2), x+2 /(n1/2))
95% confidence interval for µ.
In only 5% of all samples, the sample mean x is not in the above interval, that is 5% of all samples give inaccurate results.
A Confidence Interval (CI) has the form:
(estimate - margin of error, estimate + margin of error)
estimate is the guess for the value of the unknown population parameter
margin of error gives how accurate the we believe our guess is
11/22/2018
Daniela Stan - CSC323

6. 95% Confidence Intervalsx1 x2 x3
1 out of 25 confidence intervals does not cover the true value of µ x4 x5
x25
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Daniela Stan - CSC323

7. Properties of Confidence Intervals
It is an interval of the form (a, b), where a and b are numbers computed from the data; its purpose is to estimate an unknown parameter with an indication of how accurate the estimate is and how confident we are that the result is correct.
It has a property called a confidence level that gives the probability that the interval covers the parameter; we use C to denote the confidence level in decimal form; for example, 95% confidence level corresponds to C=0.95.
Users can choose the confidence level, most often 90% or higher because we want to be quite sure about our conclusions.
11/22/2018
Daniela Stan - CSC323

8. CI for a population mean
Choose an SRS of size n from a population having unknown mean µ and known standard deviation .
A level .95 confidence interval for µ is
(x - 2/(n1/2), x + 2/(n1/2))
in general, a level C confidence interval for µ is
(x - z* /(n1/2), x + z* /(n1/2))
The relationship between C and z*:
P(x - z* /(n1/2)< µ <x + z*/(n1/2))=C
11/22/2018
Daniela Stan - CSC323

9. Other confidence levels
Other confidence levels are found by checking the normal table
(Table D). z*
C 90% 95% 99%
1.64 /(n1/2),
Margin of error x
90% Confidence Interval
1.96 /(n1/2), x
95 % Confidence Interval
2.57 /(n1/2), x
99 % Confidence Interval
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Daniela Stan - CSC323

10. Remarks
1. Notice the trade off between the margin of error and the confidence level. The greater the confidence you want to place in your prediction, the larger the margin of error is (and hence less informative you have to make your interval).
A C.I. gives the range of values for the unknown population parameter that are plausible, in the light of the observed sample parameter. The confidence level says how plausible.
A C.I. is defined for the population parameter, NOT the sample statistic.
The confidence intervals are approximate and holds in large samples. This is because they are defined using the normal approximation.
To make a margin of error smaller, you can
take a larger sample
decrease the population standard deviation
decrease the confidence level C
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11. More on CI’s Problems 6.7, 6.10/page430 Choose the sample size:
In order to obtain both high confidence C and a small
desired margin error m when calculating the confidence interval of a
normal mean, the following equation should be solved:
m = z* /(n1/2)
Smaller m gives greater n
Sample size: n = (z* /m) 2
Problems 6.16, 6.17/page432
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12. Summary for C.I.’s
The formula for a C.I. is valid for estimates computed from a simple random sample. May not be valid for other types of samples.
A C.I. is used when estimating an unknown parameter from sample data.
The C.I. gives a plausible range for the unknown parameter
The C.I. is an interval for the population parameter (the true value), NOT for the sample estimate.
The C.I. is constructed from the sample and depends on the sample!
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Daniela Stan - CSC323

13. SAS procedure for the C.I. of a population average
PROC MEANS DATA = data-name N MEAN STD CLM
ALPHA=value MAXDEC = number;
VAR measurement-variable;
RUN;
Where
ALPHA=value is the (1-confidence level) value.
(Thus ALPHA=0.05 for a 95% C.I., ALPHA =0.1. for a 90% CI,
ALPHA=0.01 for a 99% C.I. )
CLM is the option for C.I.’s
MAXDEC = number defines how many decimal numbers (typically 1 to 4)
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Daniela Stan - CSC323

14. C.I. for population average
SAS Example
proc means data=dist n mean std clm
alpha=0.05 maxdec=4 ;
var x;
title “C.I. for population average";
run;
C.I. for population average
The MEANS Procedure
Analysis Variable : x
Lower 95% Upper 95%
N Mean Std Dev CL for Mean CL for Mean ________________________________________________________
________________________________________________________
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Daniela Stan - CSC323