STA 210 Lecture Notes - Lecture 6: Confidence Interval
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17 Sep 2019
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~ 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 it’s 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 don’t 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 don’t 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