Estimating with Confidence: Means and Proportions

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Estimating with ConfidenceMeans and Proportions

• POL 201
• Oct. 14, 2005

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Today, we will

• Learn to get better estimates with our confidence
  interval for means (using the t distribution)
• Learn to generate confidence intervals for
  proportions

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REVIEW CIs

• We have a sample mean but want to know where the
  population mean is. To answer this question what
  is the population mean? we construct confidence
  limits around the sample mean.
• 95 CI
• where
• and z is the critical value in a normal
  probability distribution for computing the upper
  and lower estimates. Number of standard
  deviations we need to capture a certain percent
  of cases (usually 95)

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Can we do better?

• We can get better estimates under some
  circumstances!
• z distribution (standard normal) is problematic
  when our sample size is small sampling
  distribution is non-normal
• We need some way to have our distribution change
  shape to reflect the uncertainty we have as our
  sample size changes
• That distribution is the t distribution

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Students t Distribution

• It is a standardized probability distribution
• Unlike z, t changes peakedness as the sample size
  varies, becoming bell shaped as the sample size
  increases.
• t test for computing confidence limits is
• 95 CI with a t for
  certain degrees of freedom
• Degrees of Freedom (df) control the size of the
  peak based on the sample size
• df n-1

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t-table
95
d.f. (n-1)
90
98
80
99
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Steps for C.I. using t

• Obtain the Std. Error
• Get a value of t from the t distribution
• Compute the Interval (Plug and Chug)

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Lets Practice

• Suppose we were interested in how frequently
  people vote. To study this, a researcher asks 10
  people how many times they have voted in the last
  5 Congressional Elections
• The average number of times a person in this
  sample voted was 2.7, with a standard deviation
  of 1.3

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• Step 1 Obtain the Std. Error

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Step 2 Get a value of t from the t
distribution D.F. n-1 9 Choose a level of
risk (.05) t critical value 2.262
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• Step 3 Compute the Interval
• In repeated samples of the same size from the
  same population, 95 of samples would yield an
  interval that contains the true mean.

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Now You Try

• A man drives 30 miles to work every day. There
  are many stop lights on the way, so it seems to
  take a different length of time each day. He
  wants to estimate the average drive time. He
  times his drive over 25 days and finds a mean
  drive time of 38 minutes with a standard
  deviation of 9 minutes.
• Using 95 level of confidence, estimate the
  average drive time with a confidence interval.

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• Step 1 Obtain the Std. Error
• Step 2 Get a value of t from the t table
• Step 3 Calculate the Confidence interval

df 24 a .05 t 2.064
38 2.0641.84 38 3.79
We are 95 confident that the True mean is
between 34.21 41.79
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Should I use z or t ?

• With t, you get more accurate results for smaller
  sample sizes.
• As the degrees of freedom get larger and larger,
  the t distribution turns into the standard normal
  distribution (the z distribution)
• As a result, we should always use t.
• Why did I have to learn this stupid z thing?

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What if I dont have a mean

• Percentage of people who vote for Bush
• Proportion of the population who is in a certain
  category
• We need another test

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Surveys and experiments often produce counts
which we can turn into proportions. Count f /
n proportion 100/600 .60 Or multiply by 100
to get a percentage 6010060
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Sampling Error for Proportions

• Proportion in a sample is not the same as the
  True population proportion.
• We can estimate Confidence Intervals for
  proportions just like means

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The Formulas

• CI for Means
• CI

• CI for Proportions
• CI

The Difference
Proportions
Means
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p proportion 1- p Not p, sometimes called
q Then proportion of people favoring abortion is
p The number of people opposing abortion is 1
p. If the sample size (n) is large enough
the sampling distribution will be normal. The
sampling distribution from which you are drawing
your one sample will approximate a normal
probability distribution a Z distribution when
Np gt 10 and n(1 P) gt 10 If nplt10 or
n(1-P)lt10, then we must use something else. We
will not encounter that this semester
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Steps for Computing a Confidence Interval for
Proportions

• Convert the frequency count in your sample into
  proportions. P count / sample size, f/n
  600/1000 .60
• Find the appropriate critical value of z.
• Use the last line on the t table for infinite
  degrees of freedom (90 1.645, 95 1.96, etc)

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• 3. Calculate the Standard Error
• 4. Plug and Chug

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Practice Problem
Would a majority of all park visitors favor
stricter controls on animals (requiring a leash)?
Can you be 95 confident that more than half the
visitors would approve stricter limits. Results
of the survey were 89 of 150 visitors favored
stricter restrictions. Step 1 Generate
Proportion 89/150 P 0.593 or 59.3
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Step 2 Find the Critical Value of Z. (Z table
or bottom line of t table). This works out to be
1.96 Step 3 Compute the Standard Error Step 4
Plug and Chug
.593 1.96.040 .593 .078 Interval is .515
to .671 95 confident that most visitors favor
restrictions
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A National SRS poll of n500 finds that 330 in
the sample favor stronger gun controls. Stated
in percent, 66 of this sample favors stronger
gun controls.
Step 1 What is the problem? What is the
percent of people in the population favoring gun
control. Convert the frequency into a proportion
330/500.66
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Step 2 Find the critical Value of Z . If we
choose 95 confidence, we use 1.96 Step 3
Compute the Std. Err. Step 4 Plug and
Chug 95 CI .66 1.96(.02) .66 .0392
The mean support for gun control is 66 3.92

.02
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As the SRS becomes bigger the estimated error
around the measure of central tendency gets
smaller. The larger the sample the less the
chance of getting an atypical average.
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Choosing between t and z for Confidence Intervals

• Proportions Always use z
• Means
• Always use t
• Why? If the sample size is large enough to use
  z, then the t table will give you the right
  value anyways.