Download Confidence Intervals

STAT 3120
Statistical Methods I
Lecture 3
Confidence Intervals for
Parameters
STAT3120 - Confidence Intervals
These notes will guide you through estimating parameter
(mean) confidence intervals. Including:
• CIs for one population mean
• CIs for the population mean of paired differences
• CIs for the difference between two population means
In each case:
1. The formula will be presented;
2. The formula will be applied (manually);
3. The formula will be applied via SAS.
STAT3120 - Confidence Intervals
As we saw previously, any CI can be estimated using
the approach of
Sample estimate + conf. level * standard error
A Confidence Interval around a single population
parameter is developed using:
x  t * (s/SQRT(n))
Where:
x = sample mean
t/2 = the appropriate two sided t-stat, based upon desired
confidence
s = sample standard deviation
n = number of elements in sample
STAT3120 - Confidence Intervals
And, again, as we saw with the proportion CIs…
Typical t statistics used in CI Estimation:
90% confidence = 1.645
95% confidence = 1.96
98% confidence = 2.33
99% confidence = 2.575
Note that these are the same as the z-scores from the
previous notes. If we have sufficiently large samples –
the t-statistics and the z-scores will be the same values.
For precise t-stat values for smaller samples, refer to the
table of t-stats in your book.
STAT3120 - Confidence Intervals
For example, lets say that we took a poll of 100 KSU
students and determined that they spent an average of
$225 on books in a semester with a std dev of $50.
Report the 95% confidence interval for the expenditure
on books for ALL KSU students.
STAT3120 - Confidence Intervals
Now, assuming that you need to maintain this MOE, but at a
99% confidence, what is the new sample size?
You can do the algebra yourself, or use the following
transformation of the formula:
n=(t)2*δ2/E2
Where:
n=sample size
t = t-stat associated with selected alpha
δ = standard deviation (of sample or population)
E = Maximum Margin of Error/Width of interval
Confidence Intervals - Software
From the PennState1 dataset, determine the 95%
Confidence Interval for the fastest speed that students
have driven.
Replicate this result manually.
Confidence Intervals
One general note regarding Confidence Intervals…
The results tell us NOTHING about the probability of an
individual observation…a 95% interval SHOULD NOT be
interpreted as “Joe has a 95% probability of having
driven between x and y MPH”.
The interval is an estimation of the mean of the
population…not of an individual.
Confidence Intervals – Pop Mean of Paired
Differences
As we saw previously, any CI can be estimated using
the approach of
Sample estimate + conf. level * standard error
A Confidence Interval around the population mean of
paired differences :
xd  t* (sd/SQRT(n))
Where:
x = sample mean (difference of the two means)
t = the appropriate two sided t- statistic, based upon desired
confidence
s = sample standard deviation (difference)
n = number of elements in sample
Confidence Intervals – Pop Mean of Paired
Differences
A few notes about paired differences (which are VERY
difference from two sample differences):
• The same (or VERY similar) people/objects are
measured pre/post treatment;
• Typically, we are only interested in the calculated
differences between the before and after - not in the
actual values of the original data which was collected.
• For reasons which will be discussed later, it is
preferable to use Paired Difference tests rather than
Independent Sample Tests – since we lose fewer
degrees of freedom.
STAT3120 - Confidence Intervals
For example, lets say that a particular firm tracks their
sales every week over the course of a year. They
average 150 units a week. After hiring an advertising
company, the average goes up to 165 units on average
the next year. The std of the differences between the
two years is 10.25. What is the 90% Confidence Interval?
Confidence Intervals - Software
From the URPDATABASE, determine the 95% Confidence
Interval for the PreUISS and POSTUISS for women…who
have had surgery…then do it again for the women who
did not have surgery.
Question – what does it mean when “0” is inside the
interval?
Confidence Intervals – Differences
between two ind. samples
As we saw previously, any CI can be estimated using
the approach of
Sample estimate + conf. level * standard error
A Confidence Interval around the difference between
two independent samples can be calculated as:
X1 – X2  t* SQRT((s21/n1)+(s22/n2))
Where:
x = sample mean (two independent samples)
t = the appropriate two sided t-statistic, based upon desired
confidence
s = sample standard deviation (two independent samples)
n = number of elements in each sample
Confidence Intervals – Differences
between two ind. samples
A few notes about independent sample differences:
• The two samples must be statistically independent of
each other – how would you prove that?
• You need to know if the variances (std) are
approximately equal or not. Without any information,
you should assume that they are not – this is a more
conservative approach. The formula from the previous
slide assumes that they are not equal.
Confidence Intervals - Software
From the CEOdata, place the CEOs into two random
groups…generate a 99% CI for their salary. Then, place
the CEOs into two age categories – less than 50 and
older than 50…generate the same interval.
Question – what does it mean when “0” is inside the
interval?