Transcript 7.2 Day 1: Mean & Variance of Random Variables

10.1 DAY 2: Confidence
Intervals – The Basics
How Confidence Intervals Behave
We select the confidence interval, and the
margin of error follows…
We strive for HIGH confidence and a
SMALL margin of error.
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HIGH confidence says that our method almost
always gives correct answers.
SMALL margin of error says that we have
pinned down the parameter quite precisely.
How Confidence Intervals Behave
Consider margin of error…

z

n
The margin of error gets smaller when…
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z gets smaller. To accept a smaller margin of error,
you must be willing to accept lower confidence.
σ gets smaller. The standard deviation σ measures
the variation in the population.
n gets larger. We must take four times as many
observations in order to cut the margin of error in
half.
Ex 1: Video Screen Tension – Part
2
Suppose the manufacturer (from yesterday’s
example) wants 99% confidence rather than
90%. The critical value for 99% confidence is
z = 2.57. The 99% confidence interval for μ
Demanding
99%
based
on a SRS
of 20 video terminals with
confidence instead

mean
x confidence
= 306.3 is:
xz

of 90%
has increased the
margin of error from
15.8 to 24.7.
n
43
306.3  2.57

20
306.3  24.7  (281.6 , 331.0)
Sample Size for a Desired Margin
of Error

To determine the sample size that will yield
a confidence interval for a population mean
with a specified margin of error, set the
expression for the margin of error to be
less than or equal to m and solve for n:
z

n
m
Ex 2: How Many Monkeys?
Researchers would like to
estimate the mean cholesterol
level μ of a particular variety of
monkey that is often used in lab
experiments.
We mustThey
roundwould like their
estimate
toWe
beneed
within
up!!!
97 1 mg/dl of the
true value
of μ attoa 95%
monkeys
confidence
level.theA previous study
estimate
indicated
that σ levels
= 5 mg/dl.
cholesterol
Obtaining
is timeto ourmonkeys
satisfaction.
consuming and expensive, so
researchers want to know the
minimum number of monkeys they
will need to generate a satisfactory
estimate.
z

n
1
5
1.96
1
n
(1.96)(5)  n
9 .8  n
96.04  n
Ex 3: 2004 Election
A poll taken immediately before the 2004
election showed that 51% of the sample
intended to vote for John Kerry. The
polling organization announced that they
were 95% confident that the sample result
was within + 2 points of the true percent of
all voters who favored Kerry.
Ex 3: 2004 Election
Explain in plain language to someone who knows no
statistics what “95% confident” means in this
announcement.
The method captures the unknown parameter 95% of
the time.
The poll showed Kerry leading. Yet the organization
said the election was too close to call. Explain.
Since the margin of error was 2%, the true value of p
could be as low as 49%. Thus, the confidence
interval contains some values of p, which suggests
that Bush will win.
Ex 3: 2004 Election
On hearing the poll, a politician asked, “What is the
probability that over half the voters prefer Kerry?” A
statistician replied that this question can’t be
answered from the poll results, and that it doesn’t
even make sense to talk about such a probability.
Explain.
First, the proportion of voters who favor Kerry is not
random – either a majority favors Kerry or they don’t.
Discussing probabilities has little meaning: the
“probability” the politician asked about is either 1 or 0.
Some Cautions
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The size of the sample determines margin
of error. The size of the population does
not influence the sample size.
The data must be a SRS from the
population.
Different methods are needed for different
designs (other than a SRS).
There is NO correct method for inference
from data haphazardly collected with bias
of unknown size.
…More Cautions
Outliers can distort results.
The shape of the population distribution
matters.
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When n>15, the confidence level is not greatly
disturbed by non-Normal populations unless
extreme outliers or quite strong skewness are
present.
So far, we have been given the standard
deviation σ of the population. We will learn
how to proceed with an unknown σ later.