Transcript Section 11.1 Third Day

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Warm up
The mean salt content of a certain type of
potato chips is supposed to be 2.0mg. The salt
content of these chips varies normally with
standard deviation σ = 0.1mg. An inspector
takes a sample of 50 chips and finds the mean
salt content of these chips to be 1.97mg. The
inspector should reject the entire batch if the
sample mean salt content is significantly
different from 2mg at the 5% significance level.
-Should this batch be rejected?
-Describe a type 1 and type 2 error.
-What is the probability of a type 1 error?
Statistics
Descriptive
Statistics
Inferential
Statistics
Confidence
Intervals
One-Sided
Confidence
Intervals
Hypothesis
T ests
There is a relationship
between
Confidence T wo-S ided
T wo-S ided
Intervals
and
Confidence
Hypothesis
Intervals Tests
Hypothesis
T ests
One-Sided
Hypothesis
T ests
How to use Table B
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If you have a two-sided graph and want
an α of .05, what is our cut off z-score?
How does this relate to Table B.
However, if your graph is two sided,
then both tails could not exceed α, so
you would look at 1- α to find your cut
off z-score.
Duality
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A level α two-sided significance test
rejects a hypothesis H0: μ = μ0 when
the value μ0 falls outside a level 1 – α
confidence interval for μ.
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Ex. If your α = .1, then you reject your
null hypothesis if your μ0 falls outside of a
90% confidence interval
Special note: we are talking about TWOsided hypothesis tests (μ ≠ μ0)!!
Some cautions
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This information is for you to
understand a link between the two.
However, if you just use a confidence
interval, you do not have a physical pvalue calculated, so you can not
complete your sentence.
You may very well see this type of
relationship on your AP exam.
Section 9.3 2nd Day
Matched pairs t-procedures
And
Robustness for t-procedures
Comparative studies vs.
single-sample investigations
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Comparative studies give us more
information and are more convincing
than single-sample investigations.
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Why?
What is a comparative study?
Matched Pairs t-procedures
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To compare the responses to two
treatments in a matched pairs design,
just subtract the data to make one list
(if you’re given all of the data).
Perform the t-procedures on the list of
the differences.
Stating Ho and Ha.
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When conducting a matched-pairs ttest, you have to be careful when you
state Ho and Ha.
In a matched pairs t-test, the null and
alternate hypotheses should have a
subscript of a D to indicate that you’re
subtracting. Then you have to define
which order you subtracted.
Matched-Pairs Design
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Recall from Chapter 5 what a matched
pairs design is.
I know we have actually already studied
a matched pairs procedure or two.
Remember investigating whether the
diet cola loses its sweetness?
Let’s look at our homework
example from yesterday
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We hear that listening to Mozart improves
student’s performance on tests. Perhaps
pleasant odors have a similar effect. To test
this idea, 21 subjects worked a paper-andpencil maze while wearing a mask. The mask
was either unscented or carried a floral scent.
The response variable is their average time
on three trials. Each subject worked the
maze with both masks, in a random order.
The randomization is important is important
because subjects tend to improve their times
as they work a maze repeatedly.
How the hypotheses are
written
Ho: μD=0,
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where μD=unscented – scented
Ha: μD > 0
Which symbol (< or >) should go
here if the claim is that pleasant
odors would improve the time it
takes to complete a maze?
Try this one on your own.
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Read the problem on page 581. Use
the data on this page.
Define what difference will be here.
Conduct a 99% confidence interval on
the difference in pressure lost.
Is there evidence that nitrogen reduces
the tire pressure loss in tires?
Assumptions
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Recall the assumptions for the t-procedures (t-intervals AND ttests)
The sample is a SRS from the population of interest
Observations from the population are distributed normally. We
don’t know this so we verify whether this is plausible.
 Is n large? If it is, then the CLT verifies that the distribution
of x-bar is approximately normal.
 If n is not large, then we need to examine the data. If the
histogram or boxplot of the data is symmetric with no
outliers (or the normal probability plot is linear), it is
plausible to assume that the data are from a normal
population.
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If the question stem tells us that the population is
distributed normally, then we do not need to verify the
assumption.
Robust?
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A confidence interval or hypothesis test
is called robust if the confidence level or
p-value doesn’t change very much
when the assumptions of a procedure
are violated.
Are the t-procedures robust?
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In reality, very few populations are
exactly normal. How does that affect
our t-procedures?
The t-procedures are actually quite
robust against non-normality of the
population IF there are no outliers.
What do I mean by “check the
data?”
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Always make a plot (boxplot, histogram,
normal probability plot) to check for
skewness and outliers.
If n is large, this step isn’t AS
necessary. WHY?
Because the CLT theorem tells us that x-bar is
distributed approximately normally if n is large.
Some practical guidelines
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Except in the case of small samples, the
assumption that the data are an SRS
from the population of interest is more
important than the assumption that
the population distribution is normal.
Homework
Chapter 9
# 80, 82, 89, 92, 93