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

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1 Oct 2019

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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

“sigma” here)

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