Transcript Section 10.2 Significance Tests


We looked at screen tension and learned
that when we measured the screen tension
of 20 screens that the mean of the sample
was 306.3. We know the standard
deviation is 43.
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Find an 80% confidence interval for μ.
Find a 99.9% confidence interval for μ.
How large a sample would you need to produce
a 95% confidence interval with a margin of error
no more than 3?
Section 9.1
Introduction to
Significance Tests
A Rose By Any Other Name

Significance Tests go by a couple of other
names:
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Tests of significance
Hypothesis Tests
Inference
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So far, we’ve learned one inferential method:
confidence intervals. Confidence intervals are
appropriate when we’re trying to estimate the
value of a parameter.
Today, we’ll investigate hypothesis tests, a
second type of statistical inference. Hypothesis
tests measure how much evidence we have for
or against a claim.
.

A significance test is a formal procedure
for comparing observed data with a claim
(also called a hypothesis) whose truth we
want to assess. The claim is a statement
about a parameter, like the population
proportion p or the population mean µ. We
express the results of a significance test in
terms of a probability that measures how
well the data and the claim agree.
The Reasoning Behind Tests of
Significance
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I say I am an 80% free throw shooter. You say
“PROVE IT!”
So, I shoot 25 free throws and only make 16. You
say that I’m a liar.
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Your reasoning is based on how often I would only make
16 or fewer free throws if I am indeed an 80% free throw
shooter. What is this probability?
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In fact, this probability is 0.0468. The small probability of
this happening convinces you that my claim was false.
Diet Colas
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Diet colas use artificial sweeteners, which lose their
sweetness over time. Manufacturers test new colas for
loss of sweetness before marketing them. Trained
testers sip the drink and rate the sweetness on a scale
from 1 to 10 (with 10 being the sweetest). The cola is
then stored, and the testers test the colas for sweetness
after the storage. The data are the differences (before
storage – after storage), so bigger numbers represent a
greater loss of sweetness.
This is a matched pairs experiment!
The Data
2.0
0.4
2.0
-0.4
2.2
-1.3
1.2
1.1
0.7
2.3
Most of the numbers are positive, so most testers found a loss
of sweetness. But the losses are small, and two of the testers
found a GAIN in sweetness. So… do these data give good
evidence that the cola lost sweetness in storage? Start by
finding x-bar.
Here’s our question:
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The sample mean is 1.02. That’s not a
large loss. Ten different testers would
likely give different sample results.
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Does the sample mean of 1.02 reflect a REAL
loss of sweetness? OR
Could we easily get the outcome of 1.02 just by
chance?
Hypotheses
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We will structure our test around two hypotheses
about the PARAMETER in question (in this case,
the parameter is μ, the true mean loss of
sweetness for this cola.)
The two hypotheses are:
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the null hypothesis (no effect or no change) represented
by H0 (H-naught)
the alternative hypothesis (the effect we suspect is true)
represented by Ha.
For the Cola problem
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In words, what is the null hypothesis?
In words, what is the alternative
hypothesis?
Our Hypotheses in symbols

In this example,
H0 :   0
Ha :   0
What is a p-value?
We’ve found p-values before.
 A p-value are
is the the
probability
getting
a
P-values
areaof of
the
sample result as extreme or more extreme
shaded
region
in our normal
given that
the null hypothesis
is true.
 The smaller
p-valuearea
is, thehas
more a
curve.
Nowthethat
evidence we have in favor of Ha, and against
name!!!
Ho.
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If the p-value is low (standard is less than .05),
reject the Ho!
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When the p-value is low, we say the results are statistically
significant.
Back to the cola
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Find the P-value for the problem. Note: We know
that the standard deviation is 1.
We find that the P-value is 0.0006.
This means that only 6/10,000 trials would result
in a mean sweetness loss of 1.02 IF the true
mean is zero. Since this is so unlikely to happen,
you have good evidence that the true mean is
greater than zero.
Types of Alternate Hypotheses
and Their Graphs
Ha :   c
This is called a
two-sided
alternative. You
shade “two sides”.
It could be less
than or greater
than.
Ha :   c
Ha :   c
These are
called onesided
alternatives.
You only shade
“one side”. It
is either
greater than
or less than.
Another example
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Cobra Cheese Company buys milk from several
suppliers. Cobra suspects that some producers are
adding water to their milk to increase their profits.
Excess water can be detected by measuring the
freezing point of the milk. The freezing temperature of
natural milk varies normally, with mean μ = -0.545°C
and standard deviation σ = 0.008°C. Added water
raises the freezing temperature toward 0°C, the freezing
point of water. Cobra’s laboratory manager measures
the freezing temperature of five consecutive lots of milk
from one producer. The mean measurement is x-bar =
-0.538°C. Is this good evidence that the producer is
adding water to the milk?
Homework
Chapter 9
#1-4, 14, 16, 17