Download Chapter 11 Powerpoint

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Recall from Chapter 10, our standardized test
statistic z.
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Remember why it’s called the standardized test
statistic. It’s because we take our test statistic x and
convert it to a z-score
z
x  o

n
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Remember the origin of our sampling
distribution: It was what you get when you
take all possible samples of a certain size,
measure the mean of each sample, and place
it in the forest of similar means. That gave
you the x distribution, whose mean  0 was
the same as our null hypothesis and whose
standard deviation was 
n
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The problem is…nobody really knows σ.
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William Gosset figured a work-around for not
knowing σ. By basing the standardized test
statistic not on z, but on a family of curves
called t, it became possible to use s, the
sample standard deviation, for inference
procedures.
The t-curves are based on “degrees of
freedom,” which is n-1. (See tables inside the
back cover of your textbook.
What we used to call
standard deviation of the
sampling distribution,
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We now call Standard Error.
s is sample standard deviation.
s
n
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This is not the last version of Standard
Error we’ll see, but every time our
measure of spread is based on sample
data, it will be called Standard Error.
n
x  o
t
s
n
It functions just like z did in Chapter 10, only now,
use the tcdf function instead of normalcdf to get
your p-value.
Tcdf arguments: (left boundary of shaded area,
right boundary, degrees of freedom)
s
x t*
n
The structure is just as we saw in Chapter 10. Get
t* from the t-table inside the back cover of your
book. In the table cross Confidence Level with
Degrees of Freedom (which = n-1)
Get s from 1-var stats using your data.
When the time comes (Step 3 of a toolbox), you
can use “T-Interval” under Stat—Test in your
calculator.
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See page 618 in YMS.
Notice that the t curves are always thicker in
the tails and shorter in the middle
Notice also that as degrees of freedom gets
large, the t curve approaches the z curve.
That means that as sample size gets larger
and larger, you may as well use z!
There another implication: The t statistic
works even with small sample sizes.
1.SRS
2.Population must be normal
(Notice—in Chapter 10, it was the sampling distribution that
had to be normal. Now it’s the population)
SRS is the more important condition.
“Population normal” is actually quite noncritical.
“Population Normal”—Relax! Of course you don’t
really know whether the population is normal, so
instead you look for evidence of non-normality.
Even then—the procedures are remarkably
forgiving. See page 636.
• If n<15, use t if data are close to normal, and there are no
outliers. (Use NPP, histogram or stemplot)
• If n at least 15, use t unless there are outliers or strong skew
in the data
• If n is big (40 or greater) you can safely use t even with clearly
skewed data.
Think Chapter 5—Experimental Design.
Remember the Random Comparative design?
Group 1
All Participants
Random
Assignment
Compare Average Response
Group 2
Group 1
All Participants
Random
Assignment
Compare Average Response
Group 2
What’s our Random Variable here?
What would equal 0 if our null hypothesis is true?
What’s our Null Hypothesis????
OMG!!!! All These Questions!!!
Group 1
All Participants
Compare Average Response
Random
Assignment
Group 2
What’s our Random Variable here?
ANSWER: The difference of the two means!
What would equal 0 if our null hypothesis is true?
ANSWER: The difference of the two means!
What’s our Null Hypothesis?
ANSWER: There is no difference between the two means!!
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Our Null Hypothesis now is: There is no
difference between the averages of the two
groups: 1  2
(or 1  2  0
)
Alternate Hypotheses can be any of these
1   2
1   2
1   2
(Note that any of these could be expressed as “the difference is
greater than zero, less than zero, or not equal to zero.”)
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Combine two normal sampling distributions
through subtraction
What’s the mean?
What’s the standard deviation?
Take a peek at the equation sheet. Anything
seem familiar?
What the heck is it, and when do you use it?
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Pooling is the idea of considering both
populations in a two-sample test really to be
part of the same population.
In other words, if Ho is is 1  2 then we’re
also asserting that  1   2
The two populations have the same variance,
in other words.
In this situation, you can use df = n1+n2 – 2,
and the “pooled” choice in your calculator.
(Reference: AMSCO p. 209)
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In other words, if Ho is 1  2  some other
number, then the the two populations are by
definition different.
Assume different variance, or “unpooled.”
Use df = the lesser of n1 - 1 or n2 – 1.
Note: your textbook neglects this idea, but
AMSCO is better on this point.