Download Section 8.1 First Day Intro to CI and Confidence Levels

Section 8.1
Introduction to Confidence Intervals
Suppose you want to guess my dog’s
weight. Which guess is most likely to
be correct?
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
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A)
B)
C)
D)
93.751 pounds
93 pounds
90  3 pounds
90  6 pounds
If we want to estimate the value of something
unknown, an interval gives us a greater likelihood of
being right.
Our intervals will be in the format
estimate ± margin of error . This is called a
CONFIDENCE INTERVAL.
We want the perfect balance between wide enough
to be fairly accurate but not so wide that there is no
meaning to the answer.
Statistical Inference

Infer – draw a conclusion
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
We’ve been doing this! When we want
to know something about a population,
what do we do?
Now, we want to use probability to
express the strength of our
conclusions. That’s where the
“confidence” in Confidence Intervals
comes from.
SAT Math
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We want to estimate the mean SAT Math
score for all California high school seniors.
Let’s suppose σ = 100 and µ is 458.
We take a SRS of 500 seniors and get an xbar of 461. Describe the shape, center, and
spread of the sampling distribution of x-bar.
Sketch the sampling distribution of x-bar.
The 68-95-99.7 Rule Revisited

According to the Empirical Rule, about
95% of all x-bar values lie within a
distance m of the mean of the sampling
distribution. What is m? Shade the
region on the axis of your sketch that is
within m of the mean.
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Since our standard deviation is 4.5, 95% of all
samples will produce an x-bar that is within 9
points of μ.
Our x-bar was 461. We say that we are 95%
confident that the true mean math SAT score
for California high school seniors is between
452 and 470.
Correct Confidence Interval Statements
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Here’s how confidence intervals work.
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We either capture the true mean in our
interval, or we don’t. For a 95% confidence
interval, we’ll capture the true mean 95% of
the time.
So, another correct way to state a
confidence interval is we are 95%
confident that the interval (452, 470)
captures the true mean math SAT score
for California high school seniors.
Illustration
Let’s start now
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You need to know the difference
between a confidence INTERVAL
and a confidence LEVEL.
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
A confidence interval is calculated from
sample data. It is of the form
estimate ± margin of error.
A confidence level gives us the success
rate for the method i.e. 95% confident
CI Construction

We want to be able to construct a CI for any
level of confidence, so we use this formula
for a CI.
statistic  critical value standard deviation of the statistic
The critical value changes based on the
level of confidence. Today, I teach you
how to calculate it.
CI for p (proportions)

Here’s the formula for a CI for p:
pz
*
p(1  p)
n
p-hat is our
unbiased
Estimate of p.
Z* is called the
critical value. I’ll
teach you how to
calculate that
next.
This is the
standard deviation
of p-hat. Notice
all the hats – we
don’t really ever
know p, so we use
p-hat to estimate
it.
How to find z*
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You will be told the confidence level (i.e. 90%).
Draw a normal curve. Label the confidence level in the
MIDDLE.
Notice there is a portion in each tail that is unshaded.
 The value of one of those tails is found by
subtracting from 1 and then dividing by 2.
 Look that value up in the BODY of Table A. Make it
positive.
i.e. 90% confidence  1 - .90 = 0.10. Divide by 2 
0.05. Look up in Table A. Z* is 1.64 or 1.65.
Try these
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Find z* for 94% confidence.
Here are some common z* that you
might want to memorize:
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90% confidence  z* = 1.645
95% confidence  z* = 1.96
99% confidence  z* = 2.575