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Confidence Interval Review

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					      Chapter 7

Inference for Distributions




                              1
• The previous chapter emphasized the reasoning
  of tests and confidence intervals
• Now we emphasize statistical practice
  – we no longer assume that population standard
    deviations are known




                                                   2
     Section 7.1

Inference for the Mean of a
        Population



                              3
                   Introduction

• So far, in all our inference for the population mean ,
  we have assumed that we know .
• The sampling distribution of x-bar depends on 
   – Thus, both CI’s and significance tests depend on 
                                    x  0
               x  (z*)       ,   z
                          n          / n

• But usually we don’t know . Then what?
   – A sensible idea would be to use s, the sample standard
     deviation, as an estimate to , the population standard
     deviation.
                                                           4
                Introduction

• Even though we are primarily interested in ,
  we are now forced to first estimate 
• We know that s changes from sample to
  sample.
  – So we are adding some variability into our
    equations.



                                                 5
       Introduction
                       
• When  is known,      n   is the standard
  deviation of the sampling distribution of x-bar.
• When  is not known, it is estimated with
  the sample standard deviation s
  – Then we must estimate the standard deviation
    of x-bar by:

  – The standard error of the sample mean:
                                                 6
                Introduction
• Now, we also know that if x-bar is normal then
               x
            z     ~ N (0,1)
                n
                                    s
• But if we use the standard error,    , does
                                     n

                    x             ?
                        ~ N (0,1)
                    s n

                                                   7
                    Introduction

• Unfortunately it doesn’t. But it does follow a
  distribution called the T-distribution.
                  x
               t          ~ Tn-1
                    s/ n
• Where n-1 is the degrees of freedom
      • From now on df = degrees of freedom
   – Notice then that for each sample size there is a
     different T-distribution.
• The degrees of freedom from this t statistic
  come from the sample standard deviation s in
  the denominator of the t statistic           8
     History of the T distribution
• The T distributions were
  discovered in 1908 by William
  S. Gosset, a statistician
  working for the Guinness
  brewing company
• He published under the pen
  name “Student” because
  Guinness didn’t want
  competitors to know that they
  were gaining an industrial
  advantage from employing
  statisticians
                                               9
                                  Brilliant!
  Properties of the T-distributions
• SIMILAR (but not the same) to a Normal dist.
   – Symmetric
   – Mean = 0
   – Bell shaped
• **The spread of the tk distribution is a bit greater than
  that of the standard Normal distribution
   – This is due to the extra variability caused by substituting
     the random variable s for the fixed parameter 
• As the degrees of freedom k increase, the tk density
  curve approaches the N(0,1) curve
   – This reflects the fact that s approaches  as the sample size
     increases (Law of Large Numbers!!)
                                                               10
  Properties of the T-distributions
• The t distribution has more probability in the
  tails and less in the center than does the
  standard Normal distribution
• Table D in the back of the book gives critical
  values for the t distributions
  – For convenience, the table entries have also been
    labeled by the confidence level C (in percent) required
    for confidence intervals


                                                        11
     D Tea-table, anyone?


– Notice that it is not nearly as comprehensive
  as the standard normal table
– This makes getting p-values a little more
  difficult, but not much.




                                                  12
   The One-Sample t Confidence
            Interval
• How does using s affect confidence intervals
  for the mean  ?
• The one-sample t confidence interval is
  similar in both reasoning and computational
  detail to the z confidence interval of Chapter 6
  – Note: “One-sample” does NOT mean that we
    have a sample size of n=1

                                               13
The One-Sample t Confidence
         Interval
• Suppose that an SRS of size n is drawn from a population
  with an unknown mean  and standard deviation . A
  level C confidence interval for  is:




   – Where t* is the value for the Tn-1 density curve with area C
     between -t* and t*.
   – The margin of error is




                                                                    14
     One-sample t confidence interval

The area between the critical
values –t* and t* under the
Tn-1 curve is C.                    Tn-1 Curve




                                           Prob=(1-C)/2
   Prob=(1-C)/2



                                                   15
                     -t*        0    +t*
                 Example 1

• The following data are the amounts of vitamin
  C, measured in milligrams per 100 grams of
  blend, for a random sample of size 8 from a
  production run:

    26   31    23    22   11    22    14   31

• We want to find a 95% confidence interval for ,
  the mean vitamin C content.

                                                  16
n=8
x-bar = 22.5
s = 7.19
       s    7.19
SE x            2.54
        n     8
From Table D we find that for 95% CI t*7 = 2.365

       s                  7.19 
 x t 
      *
            22.5  2.365         (16.5, 28.5)
       n                  8 

We are 95% confident that the mean vitamin C content for this
production run is between 16.5 and 28.5 mg/100g.
                                                                17
• In this example we have given the actual
  interval (16.5, 28.5) as our answer

• Sometimes, we prefer to report the mean
  and margin of error:
     The mean vitamin C content is 22.5 mg/100g with
      a margin of error of 6.0 mg/100g.



                                                       18
                   Testing

• In tests of significance, as in confidence
  intervals, we allow for unknown  by using s
  and replacing z by t
• Let n be the sample size
                                            x
  – If σ is known and n is large then z            ~ N (0, 1)
                                            
                                                n
  – If σ is NOT known then
                                x
                             t     ~ Tn 1
                                s
                                  n                     19
      The one-sample t test
• Suppose that a SRS of size n is drawn from a
  population having unknown mean 

• To test the hypothesis H0 :  = 0 based on a
  SRS of size n, compute the one-sample t
  statistic:
                 x  0
              t
                 s n
                                                  20
         The one-sample t test
• The P-value for a test of H0 against

  – Ha :  > 0 is P(Tn-1  t)

  – Ha :  < 0 is P(Tn-1  t)

  – Ha :   0 is 2P(Tn-1  |t|)


• These P-values are exact if the population
  distribution (X) is Normal and are approximately
  correct for large n in other cases              21
               …Example 1

• Recall that n = 8, x-bar = 22.50, and s = 7.19

• Suppose we want to test whether the mean vitamin
  C content in the final product is 40

• Hypotheses:
                 H0: µ = 40
                 Ha: µ ≠ 40
                                                   22
                    …Example 1
• Test statistic:
           x  0   22 .5  40
        t                     6.88
           s n       7.2 8
• P-value:

   2P T7  6.88   2P T7  5.408 
   2P T7  5.408   2(.0005)  0.001  P  value  0.001

• Decision and Conclusion:
   – We reject H0
   – There is significant evidence to conclude that the
     mean vitamin C content for this run is not equal to 40.   23
 Two-sided alternative and CI’s

• We can use (1-)% Confidence intervals
  to test a two-sided alternative alternative
  hypothesis at the  significance level
  – If the confidence contains the null mean (the
    mean that is assumed to be true) then this is
    a plausible value and fail to reject H0.



                                                    24

				
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posted:1/17/2013
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