M33- Confidence Intervals Handout by hedongchenchen

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									  Confidence Intervals
                        Estimation


 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals   1
  Lesson Objective
    Learn how to construct a
     confidence interval estimate
     for many situations.
    L.O.P.
    Understand the meaning
     of being “95%” confident
     by using a simulation.
    Learn how confidence intervals
     are used in making decisions
     about population parameters.
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals   2
       Statistical Inference

          Generalizing from a sample
               to a population,
             by using a statistic
                  to estimate
                 a parameter.
          Goal: To make a decision.
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals   3
             Statistical Inference

 Estimation of parameter:
     1. Point estimators
     2. Confidence intervals
 Testing parameter values using:
     1. Confidence intervals
     2. p-values
     3. Critical regions.
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals   4
            Confidence Interval
     point estimate ± margin of error

         Choose the appropriate statistic
         and its corresponding m.o.e.
         based on the problem that is to
         be solved.


 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals   5
    Estimation of Parameters
  A (1-a)100% confidence interval estimate of a parameter is
                         point estimate  m.o.e.
 Population                Point Estimator                 Margin of Error
 Parameter                                             at (1-a)100% confidence
Mean, m                              x                 m.o.e. = Z α                     σ
 if s is known:                                                               2             n
Mean, m                              x                 m.o.e. = t( α                    s
 if s is unknown:                                                         2
                                                                              , n-1)
                                                                                            n
Proportion, p:                  ^  X / n,
                                
                                p
                                p                      m.o.e. = Z α                    ˆ ˆ
                                                                                       p(1-p) n
                                                                          2

Diff. of two                                                                                2
                                                                                       s12 s2
means, m1 - m2 :                 x1  x2               m.o.e. = Zα                        +
(for large sample sizes only)
                                                                          2            n1 n2
Diff. of two                                                               ˆ      ˆ    ˆ     ˆ
                                                                           p1 (1-p1 ) p2 (1-p2 )
                                 p1  p2
                                 ˆ    ˆ                m.o.e. = Zα                   +
proportions, p1 - p2 :                                              2          n1         n2
Slope of regression              b                     m.o.e. = t( α , n-2) s
 line, b :                                                          2              Equ.2
                                           where s     MSE
Mean from a
                           y  a  bx *
                           ˆ                                                             1 (x * -x)2
 regression                                            m.o.e. =t( α                    s  +
 when X = x*:                                                         2
                                                                          , n-2)
                                                                                         n Equ.2
       Estimation of Parameters
     A (1-a)100% confidence interval estimate of a parameter is
                     point estimate  m.o.e.

 Population                                     Margin of Error
 Parameter         Point Estimator
                                            at (1-a)100% confidence

Mean, m
 if s is known:
                           x                 m.o.e.  Za  s
                                                             2       n

Mean, m                    x                m.o.e.  t( a    , n 1)
                                                                     s
 if s is unknown:                                            2            n

Proportion, p:        ^  X / n,
                      
                      p
                      p                  m.o.e.  Za  p(1  p ) n
                                                       ˆ     ˆ
                                                         2
       When is the population of
     all possible X values Normal?

     Anytime the original pop.
           is Normal,                      (“exactly” for any n).

     Anytime the original pop.
           is not Normal, but
           n is BIG; (n > 30).
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals   8
    Margin of Error for 95% confidence:

                  =     1.96 l             s       n

   To get a smaller Margin of Error:
            
            


 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals   9
          Confidence Intervals
     point estimate ± margin of error
  Estimate the true mean net weight of
  16 oz. bags of Golden Flake Potato Chips
  with a 95% confidence interval.
  Data:
    s = .24 oz. (True population standard deviation.)
    Sample size = 9.
    Sample mean = 15.90 oz.                Must assume
                                            ori. pop. is
                                              Normal
    Distribution of individual bags is ______ .
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 10
                                                            s = .24 oz.
    For 95% confidence
    when s is known:                                        n = 9.
                                                            X = 15.90 oz.
      m.o.e. =



    95% confidence interval for m:
              15.90 

 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 11
   Statement in the L.O.P.
  “I am 95% confident that
  the true mean net weight of
  Golden Flake 16 oz. bags of potato chips
  falls in the interval 15.5472 to 16.2528 oz.”

A statement in L.O.P. must contain four parts:
  1. amount of confidence.
  2. the parameter being estimated in L.O.P.
  3. the population to which we generalize
      in L.O.P.
  4. the calculated interval.
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 12
     Meaning of being 95% Confident
     If we took many, many, samples
     from the same population,
     under the same conditions, and we
     constructed a 95% CI from each,
     then we would expect that
     95% of all these many, many
     different confidence intervals
     would contain the true mean,
     and 5% would not.
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 13
 Reality: We will take only ONE sample.


              l
 15.7        15.9       16.1              X-axis
- m.o.e.      X       + m.o.e.


Is the true population mean in this interval?
        I cannot tell with certainty;
        but I am 95% confident it does.
 Making a decision using a CI.
 Question of interest: Is there evidence
 that the hypothesized mean is not true,
 at the “a” level of significance?
 If the “hypothesized value” is inside the CI,
      this value may be a plausible value.
      Make a vague conclusion.
 If the “hypothesized value” is not in the CI,
       this value IS NOT a plausible value.
       Reject it! Make a strong conclusion.
       Take appropriate action!
   Confidence level                                     = 1-a           = .95

   Level of significance                                =    a          = .05




 Department of ISM, University of Alabama, 1995-2003    M33 Confidence intervals 16
         0.5                                               The “true” population
                         Population of                     mean is hypothesized
                           all possible                          to be 13.0.
p2




         0.4
                          X-bar values,
                         assuming . . . .
         0.3                                                         Conclusion:
                                                                  The hypothesis is
         0.2                                                      wrong. The “true”
                   My ONE                                          mean not 13.0!
                 sample mean.
         0.1
                                             Middle
                                              95%
         0.0                 l                 l

                    5         8         11          14       17       20    X-axis
                                                                             23
                             l
                  5.6 7.9 10.2                  X            13.0 does NOT fall in
                    My ONE                                  my confidence interval;
               Confidence Interval.                       it is not a plausible mean.

      Department of ISM, University of Alabama, 1995-2003        M33 Confidence intervals 17
                         A more likely location
         0.5
        The data         of the population.                The “true” population
       convince                                            mean is hypothesized
                                                                 to be 13.0.
p1




        me the
        0.4
         true
       0.3                                                           Conclusion:
       mean is                                                    The hypothesis is
        smaller                                                   wrong. The “true”
        0.2
       than 13.0,                                                  mean not 13.0!
        around
        0.1
          7.9!
         0.0                 l                 l

                    5         8         11          14       17       20    X-axis
                                                                             23
                             l
                     5.6    7.9 10.2            X            13.0 does NOT fall in
                                                            my confidence interval;
                                                          it is not a plausible mean.

      Department of ISM, University of Alabama, 1995-2003        M33 Confidence intervals 18
Net weight of potato chip bags
should be 16.00 oz.
FDA inspector takes a sample.
If 95% CI is, say, (15.81 to 15.95), X = 15.88
   then 16.00 is NOT in the interval.
Therefore, reject 16.00 as a plausible
value. Take action against the company.

If 95% CI is, say, (15.71 to 16.05), X = 15.88
   then 16.00 IS in the interval.
Therefore, __________________________
___________________________________
Net weight of potato chip bags
should be 16.00 oz.
FDA inspector takes a sample.
If 95% CI is, say, (16.05 to 16.15), X = 16.10
  then 16.00 is NOT in the interval.
Therefore, __________________________.
But, the FDA does not care that
the company is giving away potato chips.
The FDA would obviously _____________
against the company
     Meaning of being 95% Confident
        If we took many, many, samples
        from the same population,
        under the same conditions, and we
        constructed a 95% CI from each,
        then we would expect that
        95% of all these many, many
        different confidence intervals
        would contain the true mean,
        and 5% would not.
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 21
    Interpretation of “Margin of Error”

    A sample mean X calculated from a
    simple random sample has
    a 95% chance of being “within the
    range of the true population mean, m,
    plus and minus the margin of error.”

       True - m.o.e.                    True            True
       mean                             mean            mean + m.o.e.

    A sample mean is likely to fall in this
    interval, but it may not.
 Department of ISM, University of Alabama, 1995-2002    M32 Margin of Error   22
       A common misconception
                m  m.o.e.                                              X  m.o.e.

  This region DOES contain                                             This region
  95% of all possible X-bars.                                          DOES NOT.




                             .95

                              m                                               Xm
-4.0    -3.0   -2.0   -1.0    0.0   1.0   2.0    -4.0
                                                3.0      -3.0
                                                        4.0     -2.0   -1.0     0.0   1.0   2.0   3.0   4.0

                                            X-axis                                                X-axis
                                                                 A random x-bar

 Department of ISM, University of Alabama, 1995-2002                         M32 Margin of Error       23
      Concept questions.
      Our 95% confidence interval is
              15.7 to 16.1.       X = 15.9
                                                                   Yes or No or ?
Is our confidence interval one of
the 95%, or one of the 5%?
Does the true population mean
lie between 15.7 and 16.1?
Does the sample mean
lie between 15.7 and 16.1?
What is the probability that
m lies between 15.7 and 16.1?
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 24
      Concept questions.
      Our 95% confidence interval is
              15.7 to 16.1.       X = 15.9
                                                                   Yes or No or ?
Do 95% of the sample data lie
between 15.7 and 16.1?
Is the probability .95 that a future sample
mean will lie between 15.7 and 16.1?
Do 95% of all possible sample means
lie between m - e.m. and m + e.m.?
If the confidence level is higher,
will the interval width be wider?
 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 25
                            End of File M33




 Department of ISM, University of Alabama, 1995-2003   M33 Confidence intervals 26

								
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