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									 Chapter 24:



Comparing Means
            Comparing Two Means

 Population model           For independent random
  parameter of interest is    variables, variances add.
  the difference between  If we know the population
  the means, 1   2 .       means
                                                   12  2 2
 The statistic of interest     SD  y1  y 2       
                                                   n1    n2
  is the difference in the
                             If we estimate, using the sample
  two observed means,         means
   y1  y 2 .                         SE  y  y  
                                            1   2
                                                       s12 s2 2
                                                          
                                                     n1   n2
          Comparing Two Means

 Confidence interval is call a two-sample
  t-interval.
 The hypothesis test is called a two-
  sample t-test.
    y  y   ME
      1    2


   ME  t  SE  y  y 
           *
                   1   2
        A Sampling Distribution for the
        Difference Between Two Means
 When the conditions are    Modeled by a Student’s t-model
  met, the standardized       with a number of degrees of
  sample difference           freedom found with a special
  between the means of two    formula.
  independent groups,
                             We estimate the standard error

 t
              
     y1  y 2   1  2    with

                  
         SE y1  y 2                       
                                  SE y1  y 2 
                                                 s12 s2 2
                                                    
                                                 n1 n2
   Assumptions and Conditions

 Independence Assumption
      Randomization
         Surveys: representative random samples

         Experiments: randomized

      10% Condition
 Normal Population Assumption
      Nearly Normal Condition
         Check both samples.

         Draw pictures!

 Independent Groups Assumption
      Think about how the data were collected.
                     Two-sample t-interval

 When the conditions are met, find the confidence interval for
  the difference between means of two independent groups.
   Since the standard error of the difference is

       
   SE y1 - y 2      
                       s12 s2 2
                          
                       n1 n2
                                ,

                             
                              *
                                      
   the interval is y1 - y2  tdf  SE y1 - y 2   
 The critical value depends on the particular confidence level C
  that you specify and on the number of degrees of freedom,
  which we get from sample size and a special formula.
       Comparing Brand Name &
          Generic Batteries
 L1: Brand Name    Find the interval that is likely
 L2: Generic        with 95% confidence to
 Plot               contain the true difference
                       G   B between the mean
                     lifetime of the generic brand
                     AA batteries and the mean
                     lifetime of the brand-name
                     batteries
          Comparing Brand Name &
             Generic Batteries
 Check the conditions:
     Independent groups assumption: batteries
      manufactured by two different companies from
      separate packages should be independent.
     Randomization: the batteries were selected at
      random from those available for sale. This is not
      exactly an SRS, but a reasonably representative
      random sample. Since the batteries come in packs,
      they may not be independent. Repeat the experiment
      for several packages of batteries.
           Comparing Brand Name &
              Generic Batteries
 Check the conditions:              Histograms
      10%: the number of               Brand Name (L1)
       sampled batteries are
       certainly less than 10% of
       all AA batteries
       manufactured by the
       companies.                         Generic (L2)
      Nearly Normal condition:
       the samples are small, but
       the histograms look
       unimodal and symmetric.
            Comparing Brand Name &
               Generic Batteries
 State the sampling                     STAT TESTS 2-SampTInt
  distribution model for the
  statistic:
      Under these conditions, the
       sampling model of the
       difference in the sample means
       can be modeled by a Student’s
       t-model with about 9 degrees
       of freedom.
 Choose your method:
      We will use a two-sample
       t-interval.
           Comparing Brand Name &
              Generic Batteries
 Interpretation: tell what the confidence interval
  means
      We are 95% confident that the mean useful life of the
       generic batteries is between 2.1 minutes and 35.1 minutes
       longer than the mean useful life of the brand-name
       batteries for this task.
      If generic batteries are cheaper, there seems little reason
       not to use them. If it is more trouble or costs more to buy
       them, then you should consider whether the additional
       performance is worth it.
        Testing the Difference Between
                  Two Means
 Two-sample t-test for the difference between the means of
  two independent groups:
      The conditions for the two-sample t-test for the difference between
       the means of two independent groups are the same as for the two-
       sample t-interval.
       We test the hypothesis H O : 1   2   O , where
       the hypothesized difference is almost always 0, using the

       statistic t 
                     y   1       
                               y 2  O
                                            . The standard error is
                              
                       SE y1  y 2     
                  
       SE y1  y 2 
                               s12 s2 2
                                  
                               n1 n2
                                        .
                   Camera Price Offers

 State the null hypothesis:                  Check to plots:
      We want to know if people are               L1: Friend
       more likely to offer a different
       amount for a used camera when               L2: Stranger
       buying from a friend or a
       stranger.
      HO: The difference in mean price
       offered to friends and the mean
       price offered to strangers is zero:
        F  S  0
      HA:The difference in mean price
       is not zero:  F   S  0
                    Camera Price Offers

 Check the conditions:                  Check the conditions:
      Independent groups                     Nearly Normal condition:
       assumption: randomizing the             Histograms of the two sets
       experiment gives us                     of prices are unimodal and
       independent groups.                     symmetric.
      Randomization condition: the           L1                L2
       experiment was randomized.
       Subjects were assigned to
       treatment groups at random.
      10% condition: this is a
       randomized experiment, so
       this condition does not apply.
               Camera Price Offers

 State the sampling distribution model of the
  statistic:
     Because the conditions are satisfied, it is appropriate to
      model the sampling distribution of the difference in the
      means with a Student’s t-model.
 Choose your method.
     We will perform a two-sample t-test.
                Camera Price Offers

 Calculate:               Draw:
      STAT TESTS
          2-SampTTests
               Camera Price Offers

 Conclusion:
     The P-value tells us that if there were no difference in the
      mean prices, the difference we have observed would
      occur only 0.6% of the time. That’s too rare for most
      people to believe, so we reject the null hypothesis and
      conclude that people are likely to pay a friend for a used
      camera a different amount than they would pay a
      stranger.
     We may want to take special care not to pay too much
      when buying an item such as this from a friend.
                  Pooled t-test

 If we are willing to assume that means’
  variances are equal, we can pool the data
  from the two groups to estimate the common
  variance and make the degrees of freedom
  formula much simpler.
 We are still estimating the pooled standard
  deviation from the data, so we use Student’s
  t-model, and the test is called a pooled t-test.
     Pooled Variance t-test for the Difference
       Between Two Independent Means
 The conditions for the pooled t-test for the difference between
  two independent means are the same as for the two-sample t-
  test with the additional assumption that the variances of the
  two groups are the same.
  We test the hypothesis H O : 1   2   O , where
  the hypothesized difference is almost always 0, using the

  statistic t 
                  y  y    . The standard error is
                  1       2            O

                SE
                 pooled y  y 
                              1         2


                              s2                s2
                  
  SE pooled y1  y 2 
                               pooled

                                  n1
                                            
                                                 pooled

                                                  n2
                                                          .
     Pooled Variance t-test for the Difference
       Between Two Independent Means
 The pooled variance is:                  n1  1 s12   n2  1 s22 .
                                 pooled 
                                s2
                                              n1  1   n2  1
 When the conditions are met and the null hypothesis is true,
  this statistic follows a Student’s t-model with
   n1  1   n2  1 degrees of freedom.
                                           *
                                                                   
  The corresponding interval is y1 - y 2  tdf  SE pooled y1 - y 2 ,       
  where the critical value t * depends on the confidence level
  and is found with  n1  1   n2  1 degrees of freedom.
                           When to Pool?
 The advantage of the pooled method is greatest when the
  samples are small.
      But this is when it’s hardest to check conditions.
      When the choice between two-sample t and pooled-t methods make a
       difference (sample size is small), the test for whether the variances are
       equal hardly works at all.
 In a randomized comparative experiment, we know that each
  treatment group is a random sample from the same population.
      So each treatment group begins with the same population variance.
      In this case, assuming equal variances is the same as assuming that the
       treatment doesn’t change the variance.
      Check the conditions: Boxplots, Boxplots, Boxplots!!!
               When to Pool?

 Because the advantages of pooling are
  small, and you are allowed to pool only
  rarely – when the equal variances
  assumption is met:
  DON’T!
 It is never wrong NOT to pool!!
                   CAUTION!!!

 Watch out for paired data.
     If the samples are not independent, you cannot
      use the two-sample methods.
     Two-sample methods can only be used if the
      observations in the two groups are independent.
 Look at the plots!
     Check for outliers and non-normal distributions.
     Make and examine boxplots.

								
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