45733: lecture 9 chapter 8

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45-733 lecture 9 (chapter 8)

• Interval Estimation

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Interval estimation, intro

• There is a population we are interested in
• We are interested in variables in this pop
• Variables fully described by distribution
• Distribution summarized by a parameter
• Parameter may be calculated from census
• Parameter may be estimated from sample
• Most of statistics is about figuring out
  parameter from information in sample

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Interval estimation, intro

• Parameter may be estimated from sample
• Estimate vs. estimator
• Some estimators better than others
• Biasedness
• Efficiency
• Mean squared error

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Interval estimation, intro

• Now, suppose we have settled on an estimator
  which we think is good
• Rule turning sample to estimate
• Need a sample in order to calculate estimate
• So, now we go out and collect a sample
• Using the sample, we calculate an estimate

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Interval estimation, intro

• The topic of interval estimation
• Given a sample, estimator and therefore and
  estimate
• How much does our estimate tell us about the
  parameter we are trying to figure out?
• An interval estimate is a range of values between
  which a parameter is likely to lie

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Interval estimation, intro

• Example
• How do our salespeoples salaries compare to the
  industrys?
• Population salespeople in our industry
• Variable S, annual salary
• Parameter E(S)
• To estimate, we take a sample of n salespeople
  and find out the value of S for each

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Interval estimation, intro

• Example
• Suppose we take a sample of 9 people.
• A good estimator (unbiased, anyway) of E(S) is
  the sample mean.
• Suppose in this sample, the sample mean is 64.5
  thousand dollars

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Interval estimation, intro

• Example
• Is E(S), the true parameter, likely to be exactly
  64.5?
• Based on our sample and estimator, could E(S) be
• 60?
• 70?
• 35?

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Interval estimation, intro

• Example
• 64.5 is our best guess at the value of E(S)
• What we would like is a range of values within
  which we are pretty sure p falls
• For example I am 95 sure that E(S) falls
  between 27.6 and 101.4
• For example I am 80 sure that E(S) falls
  between 42.0 and 87.0

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What is a confidence interval?

• A confidence interval is both
• A range of values into which a parameter likely
  falls
• A likelihood that the parameter falls into that
  range
• For example
• For example I am 95 sure that E(S) falls
  between 27.6 and 101.4
• For example I am 80 sure that E(S) falls
  between 42.0 and 87.0

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What is a confidence interval?

• A confidence interval
• Depends on
• The sample used to calculate it
• The distribution of the underlying random
  variables
• The estimator it is based on
• The width of the interval depends on
• All of the above
• How certain we wish to be

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What is a confidence interval?

• Width of a confidence interval
• Consider our example again
• For example I am 95 sure that E(S) falls
  between 27.6 and 101.4
• For example I am 80 sure that E(S) falls
  between 42.0 and 87.0
• To be more sure, I must widen the confidence
  interval
• Many-handed economists

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Common types of CI

• CI for the population mean using the sample mean
• CI for the population variance using the sample
  variance
• CI for the population proportion using the sample
  proportion
• CI for the population median (other percentiles)
  using the sample median (other percentiles)

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Common types of CI

• CI for the difference in population means using
  the difference in sample means
• CI for the difference in population variances
  using the sample variances
• CI for the difference in population proportions
  using the difference in sample proportions
• CI for difference in population medians (other
  percentiles) using the difference in sample
  medians (other percentiles)

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CI for the mean of a population

• This is by far the most common type of CI to
  calculate
• We want to know E(X)?x, the population mean of a
  random variable X
• We have a random sample X1, X2,, Xn
• We have calculated
• Sample mean
• Sample standard deviation

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CI for the mean of a population

• Our best guess at the population mean is X-bar,
  the sample mean
• It is unbiased
• It can be shown (though we dont do it) that the
  sample mean is the best estimator of the
  population mean under certain circumstances
• But what is a range of reasonable values that
  E(X) could be?

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CI for the mean, N() known ?

• This is a hard problem, lets start by making
  some assumptions
• Lets assume that X is distributed Normal with
  mean ?x and variance
• Lets further assume that we know

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CI for the mean, N() known ?

• Now, from our prior discussions, we know that

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CI for the mean, N() known ?

• Now, one thing we know how to do is calculate
  probabilities about the normal

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CI for the mean, N() known ?

• What we would really like to do is calculate a
  probability like
• But this is stupid
• There are no random variables in there!
• The probability is either one or zero, and we
  dont know which!

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CI for the mean, N() known ?

• Our strategy will be to start with a probability
  we can calculate
• And try to turn it into something like what we
  want

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CI for the mean, N() known ?

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CI for the mean, N() known ?

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CI for the mean, N() known ?

• Almost everything in that formula can now be
  calculated
• We know X-bar from our estimation
• We are assuming we know the variance
• We can use the normal table to look up the Phis
• But, what about a? What is it?

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CI for the mean, N() known ?

• Now, we need to choose a width or size or for
  the confidence interval
• The width of our confidence interval represents
  how certain we want to be about our result (what
  of the time we want to be right)
• Typical choices are
• 99, 95, 90, 80
• The width of the confidence interval determines a

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CI for the mean, N() known ?

• The width of the confidence interval determines a

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CI for the mean, N() known ?

• The width of the confidence interval determines a

-a
a
0
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CI for the mean, N() known ?

• The width of the confidence interval determines a
• For example, suppose we want a 90 CI
• We want to find a so that P-altZlta0.90
• This is a so that PZlta0.95
• From the table, this is 1.96

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CI for the mean, N() known ?

• The width of the confidence interval determines a

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CI for the mean, N() known ?

• An example
• Recall our salary example
• Suppose we know that salaries are distributed
  normally with mean unknown but standard deviation
  equal to 15
• Lets calculate a 95 CI for E(S)
• P-altZlta0.95

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CI for the mean, N() known ?

• An example
• Recall our salary example
• Suppose we know that salaries are distributed
  normally with mean unknown but standard deviation
  equal to 15
• Lets calculate a 90 CI for E(S)
• P-altZlta0.90

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Interpreting a CI

• What does a CI mean?
• Is the CI a probability statement about the
  population mean?
• Suppose we say that our estimate of mean family
  income in the US is 57,000
• Suppose we calculate a 95 CI for our estimate to
  be 56,000 to 58,000
• Does this mean that there is a 95 probability
  that the true population mean is between 56K and
  58K
• NO!

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Interpreting a CI

• What does a CI mean?
• P56ltE(income)lt58 is either
• 1 if E(income) is between 56 and 58
• 0 if E(income) is not between 56 and 58
• There are NO RANDOM VARIABLES in this probability
  statement

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Interpreting a CI

• What does a CI mean?
• But our calculation of a CI looks is a
  probability statement about something

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Interpreting a CI

• What does a CI mean?
• It is a probability statement about
• X-bar
• And about the CI itself!
• The endpoints of a CI are random variables

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Interpreting a CI

• What does a CI mean?
• So, a CI is a random interval if you like
• Sometimes this random interval will contain the
  true value of the parameter
• Sometimes this random interval will not contain
  the true value of the parameter

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Interpreting a CI

• What does a CI mean?
• If you construct it properly, the (random) 95 CI
  will contain the true parameter 95 of the time
• Imagine taking 1000 separate samples from the
  same population
• Construct a 95 CI for each of the 1000 samples
• About 950 of those 1000 CIs will contain E(X)
• (picture)

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CI for the mean, N() unknown ?

• Assuming that we know the variance is a bit odd
• Lets try to drop that assumption, now

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CI for the mean, N() unknown ?

• Recall the basis for our calculation was that (if
  we knew the variance), we could get from the
  normal table

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CI for the mean, N() unknown ?

• Now, we dont know the standard error, so we lack
  one piece of info.
• But, we know how to make a good estimate of the
  standard error

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CI for the mean, N() unknown ?

• Can we calculate some probability like this

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CI for the mean, N() unknown ?

• Recall the basis for the earlier calculation
• There is a similar fact

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CI for the mean, N() unknown ?

• The t-distribution
• Also called Students t distribution
• Looks very similar to the standard normal
• Slightly higher variance than standard normal
• Has one parameter called degrees of freedom
• As the degrees of freedom rise, the variance of
  the t-distribution goes down
• As the degrees of freedom approach infinity, the
  t-distribution becomes identical to the normal

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CI for the mean, N() unknown ?

• Can we calculate some probability like this
• Yes, we just need to have a t-table like the
  normal table

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CI for the mean, N() unknown ?

• Example
• Recall our salary example
• Suppose our sample is (in thousands)55,62,43,77
  ,89,61
• We know the mean is 64.5

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CI for the mean, N() unknown ?

• Example
• Lets calculate the sample variance

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CI for the mean, N() unknown ?

• Example
• Now, lets calculate a 90 CI
• Sample mean is 64.5
• Sample standard error is 6.65
• We want a 90 CI, so we start with

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CI for the mean, unknown ?

• Often, data really do come from a normal or
  near-normal distribution
• More often, perhaps, they do not
• So, if we have data (a variable X) which is not
  normally distributed, what should we do?

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CI for the mean, unknown ?

• Recall, that the important probability
  calculation we need to be able to do is

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CI for the mean, unknown ?

• The key fact we needed to know in order to do
  this calculation is NOT the normality of X, but

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CI for the mean, unknown ?

• If, somehow, we could know that
• Even when X is not normal, then we could again do
  the calculation of a confidence interval

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CI for the mean, unknown ?

• The central limit theorem comes to the rescue.
• Recall that the CLT says

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CI for the mean, unknown ?

• A modification of the CLT also says
• So that, as long as the sample size is large, we
  can proceed as if X is distributed normally, and
  only a small error results

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CI for the mean, unknown ?

• So, we can use

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CI for the mean, unknown ?

• Example
• Problem 6 on page 290

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CI for the proportion, large n

• Another parameter we are often interested in is
  the p from a Bernoulli random variable
• An unbiased (and in some contexts, best)
  estimator for this is the sample proportion

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CI for the proportion, large n

• Recall that for large n, the sample proportion is
  distributed approximately normal
• With mean p
• With variance p(1-p)/n

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CI for the proportion, large n

• So we could base a CI on
• Oops, that requires that we already know p!

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CI for the proportion, large n

• But, we have a good estimator of p in p-hat
• We know that p-hat is equal to p in expectation
• We know that p-hats variance goes to zero as n
  goes to infinity
• So, for large n, p-hat is a very good estimator
  for p

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CI for the proportion, large n

• So, we have a right to hope that
• This turns out to be true, also

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CI for the proportion, large n

• Example pg 299, problem 22

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CI for the variance, normal X

• It is often of interest to calculate a confidence
  interval for the population variance of a
  variable
• Recall, we based confidence intervals for the
  mean on

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CI for the variance, normal X

• Well, we know a similar fact about variances from
  a normal population

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CI for the variance, normal X

• Using the chi-squared table, it is not too hard
  to calculate things like

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CI for the variance, normal X

• Because the chi-squared distribution is always
  positive, it makes no sense to pick a and a.
• But how should we choose a and b?
• Many combinations of a and b would give the
  probability equal to say 95

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CI for the variance, normal X

• The usual way of choosing a and b
• Choose a and b so that
• First, the confidence interval has the chosen
  width
• Second, so that
• (draw picture)

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CI for the variance, normal X

• Once a and b are chosen

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CI for the variance, normal X

• Example problem 32, page 300

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CI for the variance, normal X

• Some review problems
• Page 319 58
• Page 291 12 (add the variance)