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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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
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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
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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)
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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