§
4. Statistical Inference: Sampling Distribution,
Confidence Intervals, and Tests of Hypotheses
This chapter is the central part of the statistics course. It contains the two main topics:
•
confidence intervals
•
hypotheses testing
We will do these for a few types of parameters/distributions.
In addition, we will also discuss the choice of sample size and maybe a few other issues.
The remaining two chapters on the syllabus discuss other situations (linear regression
and Analysis of Variance, abbreviated ANOVA).
There is a basic idea in all these.
Once we understand the approach, the rest is
just working out (or, for now, digesting) the “gory details”.
The rest of these notes contains parts of
§§
4.1.1, 4.1.2, 4.1.3 and the intro to 4.2.
CI. Confidence intervals (abbreviated CI)
We will introduce the general approach on a case that is most straightforward.
CI.1. Estimating the mean
μ
in a normal distribution, when the standard devi-
ation
σ
is known.
We need the following:
As
. assumptions
Es
. design an experiment, find a statistics (also called “estimator”)
Ds
. figure out the distribution of the estimator (given the assumptions)
CI
. using
Ds
, and the random sample that we drew, compute the CI
Let’s take them one at the time:
1

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As. The assumptions
Let us say that we have the following situation.
We manufacture on a machine in our factory balls. We would like to estimate
the average diameter.
From previous measurements we know that the standard deviation is
σ
= 0
.
01
in. We adjusted the machine, and we would like to see what (average) diameter
the produced balls have.

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