# 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