STA 210 Lecture Notes - Lecture 6: Confidence Interval

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17 Sep 2019
Department
Course
~ Statistics Lecture #6 ~
Sampling: More than Just “Fair”
o Probability Structure
o Population {Green, white, green, green white}
o What is the proportion of green in the population?
o List all possible samples of size 2
o Ten samples of size 2
(G,W) (G,G) (G,G) (G,W) (W,G) (W,G) (W,W) (G,G) (G,W) (G,W)
o Find proportion in each sample that are green
.5, 1, 1, .5, .5, .5, 0, 1, .5, .5
o What is the average sample proportion?
0.60
That is the true proportion of greens in the population.
This is no accident and is all made possible by the SRS.
o From the Probability
o Probability sample allows some very important and general things to be said about how
statistics computed from such samples behave from sample to sample.
o This, in turn, allows us to quantify the goodness of any inference we make from our statistic
to our parameter.
o The deliverable is the margin of error and a confidence interval, which you will see in the
readings.
o In Addition
o One can know what the variance of all those statistics would be, without physically having
them in front of you.
o And what the shape of a histogram of all those statistics would look like
o How Does that Help?
o Knowing these three things:
Allows one to have a simple, middle-school level formula for quantifying how good
your estimate of the parameter is
o So its deep, but is it useful?
o It is. Not life-changing, but yes.
o Allowed statisticians to walk through amazingly deep woods
o And come out with simple formulas for things like the Margin of Error which everyone uses
o But a lot of people dont know how to interpret
o How important is sampling to making this happen?
o Without a probability sample (of which an SRS is the easiest to think about), the challenging
middle part is not necessarily true (mean, variance, shape)
o So you dont really have a bridge to a formula for a margin of error.
o That’s why the sampling is important.
Way beyond some superficial sense of fairness
o Will take a while to make the connections we need
o Focus on the Homework
o Random is a misunderstood word
Trying to pull you more fully away from colloquial uses of the word random in this
class
o Finite population reasoning like we looked at last time
Trying to tease out some additional awareness of the difference between “random
and representative
o Fact Is
Surveys always live somewhere between the unemotional and amazing
mathematics of sampling and the complications of getting data from humans
Our job is to be “aware of the gap”
o Critical Distinction Random vs. Representative
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