Testing Hypotheses about the Difference Between the Means of

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					Testing Hypotheses about the
Difference Between the Means of
Two Populations


          STAT Chapter 9
          METH Chapter 13


                                                        S07




Review: Hypothesis testing steps
1.     Choose the appropriate test (statistical model)
2.     Create the Null and Alternative Hypotheses
3.     State the criteria for rejecting Null
4.     Do the calculations
5.     Make the decision: Reject Null or Not
6.     Draw Conclusions
7.     Write the results up in an APA-style format

(Steps 1, 2, & 3 constitute “Setting up the Hypothesis
    Test”)




Example:
         Recently, an endogenous brain neurotransmitter (NT)
     called Galanin has been discovered that appears to
     specifically affect one’s desire to eat foods with high fat
     content. The more of this naturally occurring NT that an
     individual possesses, the higher is his or her craving for
     high-fat foods. A drug company developed a drug that
     blocks Galanin without affecting the appetite for healthier
     (less fat) foods. They hope it will be useful for obesity.
         After approval by the FDA, one of the neuroscientists of
     the drug company conducted an experiment in which 15
     obese female volunteers were randomly selected and given
     the experimental drug for 6 months. Baseline and ending (6
     month) weights were recorded for each subject.




                                                                    1
                        Subject Baseline        Ending
The data for            Number Weight           Weight
                           1      165            145
the example:               2      143            137
                           3      175            170
                           4      135            136
                           5      148            141
                           6      155            138
                           7      158            137
                           8      140            125
                           9      172            161
                          10      164            156
                          11      178            165
                          12      182            170
                          13      190            176
                          14      169            154
                          15      157            143




  REVIEW: Hypothesis testing Step 1
     Choose the sampling distribution or statistical
     model
         First, ask what type of data: nominal or
         interval/ratio?
         Second, ask what question are you interested in
         answering: relationship or difference?

     Our answers (at the moment) are interval/ratio
     data and differences: therefore, z, t, or F tests

  (see Flowchart for choosing correct statistics)




  Review: Step 1 continued

     Next question is: How many conditions (or
     samples or groups)?
         If you have 1 sample of data, then you must
         choose between a z-test or a one-sample t-test
         If there are 2 conditions (or 2 samples of
         data), then you must choose between the two
         types of two-sample t-tests
         If there are 3 conditions/samples (or more),
         then you must choose the appropriate F-test or
         ANOVA (we’ll cover these tests later ☺)




                                                           2
One-sample hypothesis tests compare one
sample mean to a population mean, in order to
determine the likelihood that the sample is part of that
population, or whether it has been drawn from a
different population (are they statistically different?)

 z-tests:                    t-tests:
     Must know μ (mean)           Must know μ (mean) of
     and σ (standard              the population data
     deviation) of the            Use s (standard
     population data              deviation of the sample
                                  data) to estimate the
                                  standard error of the
                                  mean




Two-sample t-tests are used to
determine whether the means of two groups of
scores differ from each other to a statistically
significant degree

   Whenever there is more than 1 condition, or
   sample of data, the next question to ask is
   what type of subject design was used?
       Independent groups
       vs.
       Repeated measures




 There are 2 types of two-sample t-tests

    One is used for          One is used for these
    these designs:           designs:
        Random groups             Repeated measures
        designs                   designs
        Natural groups            Matched groups designs
        designs
                             Multiple names:
    Multiple names                Repeated measures
        Independent               Related samples
        groups                    Correlated samples
        Independent               Dependent samples
        samples                   Paired samples
                                  Matched samples




                                                            3
Hypothesis testing: Step 1
     What statistical test do we use?

        Two-       t-
        Two-sample t-test


     Which one?

        Repeated measures




Review: Hypothesis testing steps
1.   Choose the appropriate test (statistical model)
2.   Create the Null and Alternative Hypotheses
3.   State the criteria for rejecting Null
4.   Do the calculations
5.   Make the decision: Reject Null or Not
6.   Draw Conclusions
7.   Write the results up in an APA-style format

(Steps 1, 2, & 3 constitute “Setting up the Hypothesis
    Test”)




Step 2: Null and alternative
hypotheses?
     Null
        H0: µ1 = µ2
        H0: µD = 0


     Alternative
        H1: µ1 ≠ µ2
        H1: µD ≠ 0




                                                         4
  Step 3: Criteria for rejecting Null?
    Use standard choices for Psychologists:
        α = .05
        Two-tailed test


    Determine critical value of t:
        df = N – 1, where N is the number of pairs
        df = 15 – 1 = 14
        Check Appendix in your textbook for the critical
        value of t corresponding to the above info:
        tcritical = 2.145




  Step 4: Do the calculations

    Enter the data into SPSS
        CRITICAL POINT:
          If more than 1 data point is obtained from each subject (or
          when you’ve matched subjects), all the data goes on the
          same row for a given subject and each column is labeled
          If there are 2 samples of data, but each subject only gives
          1 piece of data, then you have 1 column for the data itself,
          and a second column to indicate which condition the
          subject is in




                          Subject Baseline              Ending
The data for              Number Weight                 Weight
                             1      165                  145
the example:                 2      143                  137
                             3      175                  170
                             4      135                  136
                             5      148                  141
                             6      155                  138
                             7      158                  137
                             8      140                  125
                             9      172                  161
                            10      164                  156
                            11      178                  165
                            12      182                  170
                            13      190                  176
                            14      169                  154
                            15      157                  143




                                                                         5
        Step 4: Do the calculations, cont’d.

           Double-check to make sure that the data was
           entered correctly

           Click on Analyze
               Then Compare Means,
                 Then Paired-Samples t-test,
                   Click on each variable to be analyzed in
                   this test, then click on the arrow in the
                   middle of the two boxes
                      Click on OK




        SPSS output
                                      Paired Samples Statistics

                                                                                Std. Error
                                      Mean            N        Std. Deviation     Mean
              Pair     baseline      162.0667             15        16.10442     4.15814
              1        endweight     150.2667             15        15.42478     3.98266


                                      Paired Samples Test

                                    Paired Differences
                                                     95% Confidence
                                                      Interval of the
                                          Std. Error    Difference
                        Mean Std. Deviation Mean Lower Upper              t     df        Sig. (2-tailed
Pair 1 baseline - endwe
                      1.80000    5.93055 1.53126 8.51577 5.08423        7.706        14           .000




        Step 5: Make the decision
              Compare the tobtained (just calculated) to the
              tcritical in order to make the decision:

                     tobtained = 7.706 and tcritical = 2.145

              Is tobtained larger than tcritical ?

              Yes. Reject the null.




                                                                                                           6
 Step 6: Draw conclusions
     There is a significant difference between baseline
     weight and ending weight.

     Do I need more info ?
          Yes.


     The average baseline weight was 162.07 and the
     average ending weight was 150.27. Therefore, the
     women on the drug lost an average of 11.8 lbs.




 You’ll need a graph:
     Click on Graphs
         Choose Bar
            Choose Simple, and at the bottom, click on
            Summaries of separate variables, then click on
            Define
               Click on both variables on the left, then click on
               the arrow, then choose OK
                  Go into Chart Editor (by double-clicking
                  somewhere on the Graph itself), and re-
                  format the graph in APA style
                  X out of Chart editor, then copy and paste
                  the graph into a word document




 Step 7: Write it up in APA-style
    A paired-samples t-test was
used to compare the
participants’ weight before
beginning (baseline) and again
after 6 months on the drug that
was purported to block
cravings for high-fat foods.
The participants lost a
significant amount of weight
(11.8 lbs) on the drug across
the six months, t(14) = 7.71, p
< .001. The data are graphed in
Figure 1.




                                                                    7
Example 2
    A psychologists is interested in determining whether
immediate memory capacity is affected by sleep loss.
Immediate memory is defined as the amount of material that can
be remembered immediately after it has been presented. Twelve
students are randomly selected from Introductory Psychology
and randomly assigned to 2 groups. One of the groups is sleep-
deprived for 24 hours before the material is presented. All
subjects in the other group receive the normal amount of sleep
(7-8 hours). The material consists of a series of slides, with
each containing nine numbers. Each slide is presented for a
short time interval (50 ms) after which the subject must recall as
many numbers as possible. The scores represent the percentage
correctly recalled.




Review: Hypothesis testing steps
1.   Choose the appropriate test (statistical model)
2.   Create the Null and Alternative Hypotheses
3.   State the criteria for rejecting Null
4.   Do the calculations
5.   Make the decision: Reject Null or Not
6.   Draw Conclusions
7.   Write the results up in an APA-style format

(Steps 1, 2, & 3 constitute “Setting up the Hypothesis
    Test”)




Hypothesis testing: Step 1
     What statistical test do we use?

        Two-       t-
        Two-sample t-test


     Which one?

        Independent samples




                                                                     8
Step 2: Null and alternative
hypotheses?

  Null
     H0: µ1 = µ2                or       X1 = X2


  Alternative
      H1: µ1 ≠ µ2               or       X1 ≠ X2




Step 3: Criteria for rejecting Null?
  Use standard choices for Psychologists:
      α = .05
      Two-tailed test


  Determine critical value of t:
      df = N1 + N2 – 2
      df = 12 – 2 = 10
      Check Appendix in your textbook for the critical
      value of t corresponding to the above info:
      tcritical = 2.228




Step 4: Do the calculations

Enter the data into SPSS
   CRITICAL POINT:
      If more than 1 data point is obtained from each subject (or when
      you’ve matched subjects), all the data goes on the same row for
      a given subject and each column is labeled
      If there are 2 samples of data, but each subject only gives 1
      piece of data, then you have 1 column for the data itself, and a
      second column to indicate which condition the subject is in




                                                                         9
    Data for                                           Subject                     Group               Percent
    Example 2:                                         Number                                          Recalled
                                                          1                            1                 68
                                                          2                            1                 73
                                                          3                            1                 72
           Group 1 =                                      4                            1                 65
           non-deprived                                   5                            1                 70
           control group                                  6                            1                 73
                                                          7                            2                 70
                                                          8                            2                 64
           Group 2 =                                      9                            2                 68
           sleep-deprived                                10                            2                 63
                                                         11                            2                 69
           group
                                                         12                            2                 66




         Step 4: Do the calculations, cont’d.
          Double-check to make sure that the data was entered
          correctly
          Choose Analyze
              Click on Compare Means
                Choose Independent-Samples t-test,
                   Put column with data in Test Variable box
                   and column with group in Grouping
                   Variable box
                     Click on Define Groups, enter correct
                     numbers
                     Then click on OK




         SPSS output
                                                      Group Statistics

                                                                                                       Std. Error
                          1=control,2=no sleep           N             Mean        Std. Deviation        Mean
                percent   1.00                                 6       70.1667           3.18852        1.30171
                          2.00                                 6       66.6667           2.80476        1.14504



                                             Independent Samples Test

                      Levene's Test for
                     Equality of Variances                              t-test for Equality of Means
                                                                                                            95% Confidence
                                                                                                             Interval of the
                                                                                      Mean     Std. Error      Difference
                          F       Sig.         t          df        Sig. (2-tailed) Difference Difference   Lower      Upper
percent Equal variances
                          .042      .842      2.019            10           .071    3.50000     1.73365     -.36282   7.36282
        assumed
        Equal variances
                                              2.019       9.840             .072    3.50000     1.73365     -.37136   7.37136
        not assumed




                                                                                                                                10
Step 5: Make the decision
  Compare the tobtained (just calculated) to the
  tcritical in order to make the decision:

  Is tobtained larger than tcritical ?

       tobtained = 2.019 vs. tcritical = 2.228

  No. Do not reject the null.




Step 6: Draw conclusions
  There is not a significant difference in percent of
  numbers remembered between subjects that were
  sleep deprived versus those that were not.

  Do I need more info ?
       Yes.


  Non-sleep-deprived subjects remembered an
  average of 70.17%, whereas sleep-deprived
  subjects remembered an average of 66.67% of the
  numbers.




You’ll want a graph of the data…
  Click on Graphs
       Choose Simple, then click on Summaries for groups of cases (at
       bottom of window), then Define
           Click on Other Statistic (e.g., mean) under Bars Represent
           box
           Then choose your data column on the left, and click on the
           arrow in the Bars Represent box to move it in as the Variable
           Click on the group column on the left, then click on the
           arrow to move it into the Category Axis box
              Click on OK
                  Go into Chart Editor and format it for APA style
                  Copy and paste it into your word document




                                                                           11
 Step 7: Write it up in APA-style
     The percentage of numbers
recalled correctly for sleep-
deprived versus non-sleep-
deprived students was analyzed
with an independent-samples t-
test. The percent recalled for
the sleep-deprived participants
did not differ significantly
(t(10) = 2.02, p = .071) from
the percent recalled by the
participants that slept the
normal amount. As can be
seen in the graph (see Figure
1), the two groups did not
differ.




 Underlying Assumptions for t-tests
     In theory:
          The data should be normally distributed within each
          population
          The variances of the two populations should be equal

     In practice:
          You can generally ignore the assumption of normally
          distributed data in the population
          You can ignore the assumption about equal variances as
          well, as long as the number of subjects in each
          condition is approximately equal




 Confidence Limits on MD
     Paired-samples t-test:
          XD ± (tcritical) ( sD)


     Where:

          MD = mean of the difference scores
          sD = standard error of the mean of the
          differences




                                                                   12
Confidence Limits on µ1 - µ2
  Independent-samples t-test

      (X1 – X2) ± (tcritical) ( s x1-x2)

             Standard error of the difference between means

 = (N1 – 1) s12 + (N2 – 1) s22 (1/N1 + 1/N2)
 √      N1 + N2 - 2




Confidence limits for the 2 examples

Example 1:                        Example 2:

11.8 ± (2.145) (1.53126)          (70.1667 – 66.6667)
=                                 ± (2.228) (1.733652)

11.8 ± 3.2846 =                   = 3.5 ± 3.86258

8.5154 – 15.0846                  = -.3626 – 7.3626




                                                              13