STA 210 Lecture Notes - Lecture 8: Confidence Interval, Sampling Distribution, Standard Deviation

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~ Statistics Lecture #8 ~
The Probability Beneath the Common Sense
09/26/19
o Confidence Interval Interpretation
o What’s wrong with “There is a 95% chance that the true proportion of all Americans (had
they been asked) who think the government is trying to do too much is between 50% and
58%”?
It attaches the idea of chance to the parameter. While the practical upshot may be
the same to lay ears, we push for a more complete understanding in here so that you
know what is bouncing around and what it not.
The randomness is associated with the confidence interval that the sample
produced.
Specifically the randomness is fully inherited by the statistic that forms the center of
that confidence interval.
o BN 2.20 Due Monday
o In a repetitive way, basically the same answer- more or less- for each of the first set of
prompts and that answer is on the previous slide!
o Based on a real study of professors and student showing just how prevalent the wrong
interpretation is. Speaks to the extent of the confusion and legitimizes the need to know
better.
o But does it really matter? Probably not so much in practice because interpretation may just
be too subtle, but it does in class.
o We might let our friends and co-workers get away with the slightly wrong interpretation.
Outside of this class we might even use the slightly wrong one ourselves. But we have to
prove in this class that we do know better.
o For Today
o Connect with the probability lurking in bell-shaped curves.
o Begin to connect to sampling distributions and how the MOE comes about.
o Works Like This
o Start with a generic rule about bell-shaped distributions (smoothed-out histograms)
Empirical Rule
o Leverage the fact that the sampling distribution of the sample proportion is a particular bell-
shaped distribution!
o Empirical Rule
o Tells you about how numbers that follow a bell shape distribution behave.
o Deceptively amazing
o Applies to all bell-shaped distributions
o The language here will be the stumbling block, if any appear, not the computations
o You saw a bell shape emerge organically in an in-class activity.
o Empirical Rule Setup
o Keys to using Empirical Rule on bell shapes:
Know or postulate the presumed value of the
place under the ball called the mean of the
distribution (denoted mu” here)
Know or estimate how spread out the bell is
in terms of “Standard Deviations(called
“sigmahere)
o Same words used (mean and standard deviation) as
in Module 1. They have a bit of a different meaning
here. Intuitive ideas embodied in both are the same,
however, so we are not worrying about difference in context.
o Apply this to Sampling Distribution of Sample Proportion
o It is bell-shaped
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