Point and Interval Estimation with Confidence

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					Point and Interval Estimation
      with Confidence

              Another Spectacular
              Power Point
Point Estimates
• Definition:
     A single number that is based on
  sample data, and represents a reasonable
  value of the characteristic of interest.
- Things Like:
     x    s     M
- The simplest approach to estimating a
  population value.
Lets Look at an Example
•   A recent survey asked     • What could we use to
    high school senior how        estimate µ?
    many hours they spent     •   M or x
    on line each day. Find    •   Find both:
    the point estimate of µ
    with the data below.      •   x = 46.25 = 4.625
    4 7.25 5 5.25 6.25                  10
    6.25 4.5 3 .5 4.25            M = 4.75
                              •   Which one is better?
                              •   Recall when it is best to
                                  use each one…
Confidence Intervals
(estimating µ)
• Things to Remember About the Distribution of x:
        1) Has a normal distribution (by CLT)
        2) Is an unbiased estimator of µ (x = µ)
        3) If permission slip is granted:
               s = σ/√n
•   Issues with Point Estimates: if we re – do the
    sample, will we get same stat?
Confidence Intervals
Def: the interval of reasonable values of the
  characteristic of interest
Pattern for the formula: (what’s on AP test)
statistic ± critical value(standard deviation of stat)

For Estimating µ: x ± z*(σ/√n)

                  Margin of Error
Confidence Levels
Def: the success rate of the method used to
  construct the interval.

Ex: A 95% confidence interval = if we took lots of samples
  and found the mean over and over again, in the long run
  95% of the resulting intervals would capture the true
  value of the parameter.

If we go through the same process with the same sample
   size, σ, and z*, but different x, 95% of those intervals
   will contain µ. (See pg. 511)
Calculating Confidence Intervals
What is z*?
  A location on the normal curve that
  results in the needed confidence level.

   Ex:                   95%
                   -z*          z*

Use Table C!
z* = 1.96
Calculating Confidence Intervals
Let’s try another z*:

                        -z*         z*

What is z*?
z* = 2.326
Critical Values
z* is also called a
  critical value. Any
  ideas why?

The area outside of the
  interval are called
  critical areas
Calculating Confidence Intervals
Let’s do one already! Gosh…
Suppose we need to verify the amount of
  active ingredient in a new drug. σ is
  known to be .0068 grams per liter. Here
  are the results of 3 measurements. Find a
  99% confidence interval for the true
      .8403     .8363      .8447
Calculating Confidence Intervals

• What we Need:
  x         z*        σ         n
 .8404 2.576         .0068      3
• Fill in the formula: x ± z*(σ/√n)
       .8404 ± 2.576(.0068/√3)
            .8404 ± .0101
            .8303 to .8505
Calculating Confidence Intervals

• Use the same problem, but do a 90% C.I.
    .8404 ± 1.645 (.0068/√3)
         .8404 ± .0065
         .8339 to .8469
• Compare the 90% to 99%
  What happened to the area? z*? M of E?
 See why we are “more” confident!
How Intervals Behave
1) Higher confidence gives up “precision.”
2) Want to have a small margin of error:
   - smaller z*: again calls for lower
   - smaller σ: smaller variation among
   - larger n: better picture of population
        AKA: Law of Large Numbers
        Need 4x sample size to cut m of e in half
Choosing a Sample Size
Usually decided by desired margin of error

Formula: m ≥ z*(σ/√n)

• Make sure to round appropriately!!
• Population still needs to be 10X as big as
Choosing a Sample Size
• Example: From previous problem, what
 sample size is required to produce results
 ±.005 with a C.I. of 95%?

 .005 ≥ 1.96 (.0068/√n)

Do some algebra! √n≥(1.96∙.0068)/.005
n ≥ 7.1 so n = 8 because 7 is not enough!
1) Data must come from a SRS! We don’t
  get into stuff for stratified, or blocked.
2) Bias is still bad! Fancy formulas cannot
  rescue badly produced data!
3) Outliers can have a great effect on
  intervals (think about why).
4) Need sample sizes of 15 or more.
5) For now we are assuming we know σ.

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