Transcript Chapter 10 notes

AP Statistics
Chapter 10 Notes
Confidence Interval
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Statistical Inference: Methods for drawing
conclusions about a population based on sample
data.
Level C Confidence Interval (2 parts)
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1. Confidence interval calculated from the data.
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Estimate ± margin of error
2. Confidence level – gives the probability that the
interval will capture the true parameter value in
repeated samples. (most often 95%)
Conditions for constructing a CI
(for μ)
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Data must come from an SRS.
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Independence: N > 10n
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Sampling distribution of
is approx Normal
Critical Values
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Values (z*) that mark off a specified area under
the Normal curve.
Confidence Interval for a Population
Mean
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Choose an SRS of size n from a population
having an unknown mean μ and known standard
deviation σ. A level C confidence interval for μ
is…
Steps for Constructing a CI
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1. Identify the population and parameter of
interest.
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2. Verify that all conditions are met.
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3. Do confidence interval calculations.
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4. Interpret the results in context.
Example
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Suppose that the standard deviation of heart
rate for all 18yr old males is 10 bpm. A random
sample of 50 18-year-old males yields a mean of
72 beats per minute.
(a) Construct and interpret a 95% confidence
interval for the mean heart rate μ.
(b) Construct and interpret a 90% CI.
(c) Construct and interpret a 99% CI.
Interpretation
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We are 95% confident that the true mean heart
rate of all 18 year old males is between 69.23
bpm and 74.77 bpm.
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What does 95% confidence mean?
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95% of the samples taken from the population will
yield an interval which contains the true population
mean heart rate.
Margin of Error
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Margin of Error gets smaller when…
z* gets smaller. (lower z* = less confident)
 σ gets smaller. (not easy to do in reality)
 n gets larger.
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Using the heart rate example, what would my
sample size need to be if I want a 95%
confidence interval with a margin of error, m,
of only 1 beat per minute?
Interval for unknown σ
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If we don’t know σ, (we usually don’t), we can
estimate σ by using s, the sample standard
deviation.
is called the standard error of the sample
mean .
Known σ  z distribution (Standard Normal)
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Never changes
Unknown σ  t distribution (t(k))
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Changes based on its degrees of freedom k = n - 1
One Sample t-interval
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A level C confidence interval for μ is
 t*
is the critical value for the t(n – 1) distribution.
Paired t Procedures
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Used to compare the responses to the two
treatments in a matched pairs design or to the
before and after measurements on the same
subjects.
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The parameter μd in a paired t procedure is the mean
difference in response.
Robust: accurate even when conditions are not
met.
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t procedures are not robust against outliers but are
robust against Non-Normality.
Confidence Interval for p
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Conditions:
SRS
 Independence: N > 10n
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and
are > 10.
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Confidence interval for unknown p.
Finding sample size
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To find the sample size needed for a desired C
and m…
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p* is a guessed value for p-hat. If you have no
educated guess, then say p* = .5.
Reminders
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The margin of error only accounts for random
sampling error. Non-response, undercoverage,
and response bias must still be considered.
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Random sampling: allows us to generalize the
results to a larger population.
Random assignment: allows us to investigate
treatment effects.
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Confidence Interval Summary
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1. State the population and the parameter.
2. Explain how each condition is/isn’t met.
(a) SRS
(b) Independence: N > 10n.
(c) Normality:
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For p: and
are > 10.
For μ: Look for large n. (Central Limit Theorem)
If n is small, look to see if the data were sampled from a
Normal population. At last resort, look at the sample data to
make sure that there are no outliers or strong skewness.
Summary Continued
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3. Calculate the confidence interval.
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Estimate ± margin of error
Summary Continued
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4. Interpret the interval in context.
We are ____% confident that the true
population mean/proportion of ____________
falls between (
,
).
If you are asked to interpret the confidence
level…
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______% of the samples taken from the population
yield an interval which contains the true population
mean/proportion.