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Lecture 13

STATS 13 Lecture Notes - Lecture 13: Central Limit Theorem, Standard Deviation, Interval Estimation


Department
Statistics
Course Code
STATS 13
Professor
Tsiang, Mike
Lecture
13

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Section 3.2 Continued!
!
Confidence intervals!
Definition!
a confidence interval is an interval or range of plausible values for the population parameter!
confidence level -- level of confidence in this interval!
radius of the confidence interval is called the margin of error!
the confidence level tells us how often out interval estimation method is successful at capturing the true
population parameter, not how often a single observed confidence interval is successful!
!
Simulating!
the true or hypothesized proportion π !
sample size n!
number of repetitions (1000) !
confidence level!
resulting confidence intervals in green contain π!
resulting confidence interval in red do not contain π!
!
Confidence level and Margin of Error!
can change confidence level by changing margin of error!
increasing margin of error increase confidence level!
decreasing margin of error decreases confidence level!
more precise interval = smaller range over population proportion!
a smaller margin of error is helpful but more likely to be wrong-- less confidence!
want a margin of error that will produce an interval that is informative and useful with a good confidence level!
!
selecting a margin of error!
when we have a large enough random sample, the central limit theorem tells us that the sample proportion will be
within about two standard deviations of the population proportion with probability .95!
by choosing two standard deviations as the margin of error, we cover the population proportion in about 95% of
our samples (2SD method)!
p
̂ ± 2SD(p
̂) is a confidence interval for π with an approximate 95% confidence level. !
approximate 95% confidence interval for π !
approximate 68% confidence interval for π is p
̂ ± 1SD(p
̂)!
approximate 95% confidence interval for π is p
̂ ± 2SD(p
̂)!
approximate 99.7% confidence interval for π is p
̂ ± 3SD(p
̂)!
confidence interval for π constructs using theory based approach is also called one sample z interval !
!
Finding balance!
decreasing margin of error from 2 to 1 standard deviation decreases confidence by 27 percentage points (95% to
68%)!
increasing the margin of error from 2 to 3 standard deviations increases confidence by 4.7 percentage points
(95% to 99.7%)!
95% confidence intervals are so common!
two standard deviations usually gives a good balance between high confidence and reasonable margins of
error!
!
Commonly used margins of error!
definition!
for a given confidence level, margin of error is calculated by!
margin of error = z*SD!
multiplier z* is the number of standard deviations to include in the margin or error!
when z* =1 the confidence level is 68%!
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