Download AP Stats Ch 10-1 and 10-2 and 10

Confidence Intervals
10.1
“Estimating a population parameter
(such as mean) based on a sample
statistic (its mean)”
The value of a point estimate
If all we did was to assume that  = x-bar, we
would be wrong 100% of the time! Close,
maybe, but never right on.
Why? It would be like asking, “What’s the
probability in a normal distribution that x = 5?”
No can do.
Consider a sampling distribution of
means, with the “true” population
mean in the middle.
x
If you were to take a sample, what’s the
probability that its mean would be
within 2 SD of the population mean?
.95, yes?
Isn’t it true that if my sample mean is within two SD’s of the “truth,” then
the “truth” is within two SD’s of my sample mean?
The Interval Estimate


Instead of estimating the pop mean using just xbar, now we’re going to say, “I can’t tell you for
certain precisely where it is, but I can tell you
that it’s likely to be within, say, two standard
deviations of my x-bar.”
Bear in mind—we’ve already established (CH 9)
that the mythical x-bar distribution has the same
mean as the population.
The Confidence Interval

Made up of two elements:

Point Estimate:
  
x  z *

 n

Margin of Error
The Confidence Interval


Think of it as a middle
(Mr. Warmingham’s
Head) with one Margin
of Error going out in
each direction (Mr.
Warmingham’s Arms)
The Confidence Interval
is two MOE’s wide!
The Confidence Interval



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Take your x-bar
Go one “Margin of Error” in
either direction.
Those values form the
confidence interval.
Example: You wish to estimate
the mean GPA of LP students
based on a sample of size 30 to
a 95% confidence level. You
know historically that the
standard deviation of this
distribution is .5. Your sample
mean GPA is 2.8.
The Confidence interval is
  
x  z *

 n
Conditions or Assumptions

Conditions for a confidence interval for
means:
1.
Data needs to come from a Simple
Random Sample
 2.
The sampling distribution for means must
be normal (or approximately normal).

The Confidence Interval
x
= 2.8
Z*
= 1.96, from the t-table
σ
= .5
n
= 30
  
x  z *

 n
Plug & Chug, and get the interval (2.62, 2.98)
Interpretation: “We are 95% confident that the true LP
mean GPA is between 2.62 and 2.98”
Good Interpretation:
“We are 90% confident that the true mean is between
6.5 and 6.8 books per backpack.”
Bad Interpretation:
“There is a 90% probability that the true mean is
between 6.5 and 6.8 books per backpack.”
This is bad because the truth either is, or isn’t within the
interval. The probability of it being in the interval, therefore,
is either 0 or 1 !
Really Bad Intepretation:
90% of all Freshmen have between 6.5 and 6.8 books
in their backpack. (Yuck!).

“If we were to repeat this experiment many
times in the same fashion, we would expect 95%
of the sample means to be within 1.96 standard
deviations of the mean.”

Q 10-1/1 #1 c) “95% confidence means that if
we were to repeat this exercise many times, we
would expect that 95 percent of the intervals
constructed this way would include the true
proportion of women who don’t have enough
time to themselves.”
The Inference Procedure Toolbox
Your constant friend in 2nd Semester 




State the population
State the parameter of interest in words and
symbols.
Name the procedure
State and satisfy conditions
Carry out the math
State your Conclusion in Context
Example: Estimate the mean number of books in backpacks
of Freshmen to a 95% confidence level. Your sample size is 32
(an SRS), and the average you found is 5.6 books. The sample
standard deviation was 1.4 books.
1.
2.
3.
Pop: Freshmen
Parameter: Mean number of books in backpacks, µ
Procedure: Confidence Interval for Means
Conditions: Simple Random Sample? Given in problem statement
Normal sampling distribution? Yes, by invoking the Central Limit Theorem
with n = 32.
Note: Estimate σ using the sample s.
CI: x  Z * 
n
1.4
32
(5.11, 6.08)
5.6  1.96
4.
Conclusion: We are 95% confident that the true mean number of books in
the backpacks of Freshmen is between 5.1 and 6.1 books.
Tests of Significance
10-2
The other type of Inference Procedure
(Along with Confidence Intervals)
“How Significant is this sample
mean we found?”

The basic idea:
An outcome that should rarely happen if a claim
were true, but that happened anyway, is strong
evidence that the claim is not true.
 We measure how rarely an outcome should happen
using what’s called a “P-value.”

The P-Value


Def: A P-Value is the probability of having found your
sample mean (or one that’s more extreme), assuming
the population mean is “where the Null Hypothesis
says it is.”
The Null Hypothesis is the notion that:





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Nothing’s going on
There’s no meaningful change
The aspirin really doesn’t reduce pain
The machine really isn’t out of tolerance
The stimulus really didn’t affect the economy
The advertisement really didn’t help sales.
Null Hypothesis and the
Scientific Method


As researchers, we must always assume that the
new treatment being investigated really doesn’t
have any effect, i.e. that the Null Hypothesis is
true, not the Alternate Hypothesis.
Only when dragged kicking and screaming by
the data (or really, a low P-value) must we reject
the Null Hypothesis in favor of an Alternate
Hypothesis. If the evidence is not sufficient, the
Null is not rejected!
Back to P-Value and how a
hypothesis test works



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1. Assume Ho is true.
2. Create a sampling distribution based on Ho
and sample size
3. Place the observed sample mean upon the
sampling distribution and calculate a P-value.
4. If that P-value is small enough, reject Ho.
In other words…

The smaller the P-value, the less likely you are to
have found such a thing…
By chance alone
2. If the Null Hypothesis is true.
The P-value we’ve got to be below is called α if we
want to reject Ho. This is called a significance level.
Say someone declares α to be .05. You produce a pvalue of .04. We’d say “Reject Ho at the .05
significance level.”
1.


Where do low P-Values come from?
Three possibilities
1.
2.
3.
Mere chance (bad research luck, in other
words—unlikely)
Bad Sampling
Wrong Ho!
By insisting on a low P-value, the first is unlikely.
By insisting on an SRS, the second is controlled.
That leaves Number 3 
“If the P is low, the Ho must go!”
“If the P is low, the Ho must go!”
A result that causes us to reject Ho we call
“statistically significant.” It’s a finding* so rare
that chance alone would very seldom produce it.
Because it happened anyway, it probably wasn’t
mere chance that caused it!
* A sample mean, for example
Example: In a test of new
tires, a tire company
measures maximum lateral
g-forces experienced by a test
car running around a
skidpad.
The historical average for tires of this type is 0.72 g, with a standard
deviation of .04 g.
During a test of 30 sets of tires on the test car, an average was .735 g. Is this
statistically significant evidence that the new tires are grippier than the old
model?
A Toolbox! Yea!

Step 1:
Population: Tires of this new type
 Parameter: Average maximum g-forces, µ
 Ho: The average of this new type is just like the old.
µ = .72
 Ha: The average for this new tire is greater than the
old. µ > .72


Step 2
Procedure: 1-sample z-test for means
 Conditions:

SRS? We must assume these 30 trials are an SRS of all
possible trips around the skidpad these tires could
experience.
 X-bar distribution normal? Yes, with a sample size of 30,
per CLT.


Step 3 (Version 1—an actual Z-test)
x  0 .735  .72

 2.054
.04
/ n
30
P( z  2.054)  normalcdf (2.054,1E 99) .02
z
Because this would happen only 2 times in 100 by
chance alone if the true mean for these tires is .72,
we reject Ho.


Step 3 (Version 2: Use the sampling distribution
directly) (See whiteboard)
Again, though…
Because this would happen only 2 times in 100 by
chance alone if the true mean for these tires is .72,
we reject Ho.

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Step 4:

Evidence from this sample suggests that the mean
lateral g’s for these new tires is significantly greater
than the historical average of .72. They are evidently
stickier tires.
Notice what’s mentioned in Step 4: CONTEXT. The population and
parameter, to be more precise. Refer to whichever hypothesis you now favor.
In this case, refer to Ha.
An unfortunate non-sequitur

Common mistake by students:

“Step 4: The tires are significantly sticker.
Therefore we reject Ho.”
Yuck! Cart before horse! Your conclusion is a
consequence of having rejected Ho—not vice versa.
And besides, never finish an inference procedure with
your reject/fail to reject statement. Finish with your
conclusion in context. (Keeps bosses happy).
Choosing α
and
Statistical vs. real
significance
10-3
Don’t blink. This goes quickly.
Choosing significance level

Consider two main things when deciding whether to go
with a more relaxed α (.05) or a more stringent α (.01,
.001).
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How much “convincing” is necessary for your purpose.
(Who’s your audience?
How expensive or costly would be the consequence of
rejecting Ho. If rejecting Ho would be costly in dollars, lives,
careers, you’d better require your findings to climb a tall
mountain. (a small α)
IRL, α level isn’t all that important. p = .048 is not that
different than p = .052.
Statistical vs. Real-life
Significance

“We’re thrilled you’ve got a small P-value, but so
what?”


Not all findings—though they reach statistical significance—
really matter.
See Example 10.18 on p. 588.
Small P-Values don’t fix bad
sampling

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Don’t shout Eureka until you’re sure your
sampling & methodology is correct.
In fact, your small p-value may be precisely
because of cheesy sampling. Sometimes your
small p-value is evidence that you messed up!
10-4
(Get tough—this one’s chewy)

Type 1 Error: Rejecting Ho when you really
shouldn’t have.

Example: You replace all the bearings in a piece of
manufacturing equipment, but afterwards the
machine is still pretty sloppy.

Example: A jury finds a defendant “not guilty.” (Ho
was “Not Guilty” and they failed to reject it). Later,
a new witness comes forth and provides evidence
that the guy really did do the crime!.
10-4

Type 2 Error: Failing to Reject Ho when you
really should have!
Example: Ho: Student doesn’t know the material
for a test. Test score is low, so teacher gives student
an F. Later it’s found that the student is strongly
dyslexic, and was unable to properly choose among
multiple-choice answers.
 Example: A jury rejects Ho (Ho was “Not Guilty”)
and convicts a defendant. Later, DNA evidence
reveals the defendant couldn’t have done the crime.
