Hypothesis testing

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					Hypothesis testing

   Research II Class 5
The logic of hypothesis testing
 Start with an example:
 We have a representative sample (400) of primary school
  children in TinSuiWai
 Average self-esteem: 30
 All primary school children in HK: 32
 Question: Is TinShuiWai students different from HK
  students?
 The logic is we assume they are equal in the first place
 Calculate the probability of sampling distribution of mean
    based on Central Limit Theorem
   See if 32 is very unlikely
   If so, reject the assumption (null hypothesis that they are
    equal)
   SE = 4/20 (0.5)
   95% between 30 +/- 1.96 x 0.5 (29.02 and 30.98)
   32 is outside 95% of cases, so very unlikely
   Reject the null hypothesis
 H0: There is no difference between the mean self-esteem
  score of all primary school children in TinShuiWai with that
  of the primary school children in the general population
 H1: There is difference between…


 OR
 H0: µ   (TinShuiWai)   = 32
 H1: µ (TinShuiWai) ≠ 32
 Where µ is the mean self-esteem score.
 Why should we use this indirect method?
 Because the null hypothesis is unique and parameters are
  fixed.
 The null hypothesis is the statistical hypothesis, which differs
  from the theoretical hypothesis
   What explain the difference between the levels of self-
    esteem among students?
   A possible answer “Class performance accounts for the
    different levels of self-esteem among students”
   This is theoretical hypothesis
   We have confidence if we have more evidence. If we find
    counter-evidence, we have to reject the hypothesis and look
    for other possible answers
   Designs to test this hypothesis
   Collect a sample of school-children in HK, divide the students two groups,
    one is above average performance in their class, another below it. Then we
    compare whether those with better class performance has a higher level of
    self-esteem than those lower.
   Based on the same sample, try to see if self-esteem scores correlate with
    performance in their class (for example, what percentile their performance is
    in class).
   Get a sample of students and randomly assign them into two groups, one of
    them were given a proven “learning performance improvement training”
    (suppose there is one), the other serves as a control group. This is a
    Randomized Clinical Trial (RCT). Then we run the training programme,
    and after the completion of the programme, we try to find out whether there
    is an improvement in their self-image (suppose we are quite certain that the
    training programme improved their class performance).
Method 1
 Divide the sample into two groups, one with higher level of
    class performance, one with less
   Use t-test
   H0: There is no difference in the average self-esteem
    score between those with better and worse class
    performance among the school children in HK
   H1: There is difference between……
   OR
   H0: µ(better class performance) =µ(worse class performance)
   H1: µ(better class performance) ≠ µ(worse class performance)
                Difference in self-esteem



                                            Theoretical hypothesis
Class performance made a difference



                             Statistical hypothesis

H0               H1
Type I error (α) is the probability of rejecting Ho when it is in fact true.
Type II error (β) is the probability of not rejecting Ho when it is in fact false.
One sample t-test

   Research question: Does the population’s trust
    towards social workers differ from the average
    level, which equal to 6 on a 0-10 scale?
   Null hypothesis: The population’s mean value of
    trust towards social workers is equal to 6
   Alternative hypothesis: The population’s mean
    value of trust towards social workers differs
    from 6.
                   One-Sample Statistics

     N             Mean         Std. Deviation     Std. Error Mean
社工           503       6.29                1.855              .083


                                 One-Sam ple Test
                                           Test Value = 6
                                                                95% Confidence Interval
                                                                   of the Difference
         t           df         Sig. (2-tailed) Mean Difference   Lower        Upper
社工   3.462                502              .001               .286   .12          .45
 If the null hypothesis is true, that is the population score
  of the perception towards social workers is 6, then the
  chances of having the current level of differences (might
  it be positive or negative) is 0.01 only.
 We are 99% confident that the sample mean differs from
  the population mean at this magnitude.
 In fact, if the population mean was 6, the value of 6.29
  will have 95% chances greater than the sampling mean of
  the population (with true population mean equal to 6)
  between 0.12 and 0.45.
Testing the differences about two
related means


   Research question: Does the population’s trust towards
    social workers differ from that of lawyers?
   Null hypothesis: There is no difference between the
    population’s mean value of trust towards social
    workers and lawyers.
   Alternative hypothesis: The population’s mean value of
    trust towards social workers differs from that of
    lawyers.
Paired Samples Statistics



                            Mean             N         Std. Deviation      Std. Error Mean

 Pair 1      社工                6.29              503              1.855                .083
             律師                5.96              503              2.023                .090


                                                          Paire d Sam ple s Te st
                                                   Pa ire d Diffe rences
                                                                        95% C onfidence Inte rval
                                                                           of the Diffe rence
                             Mea n     Std. Dev iation Std. Error Me an   Lowe r        Uppe r       t        df         Sig. (2-taile d)
   Pa ir 1   社工 - 律師           . 330          2. 392             . 107       . 120           . 540   3. 094        502            . 002
 We reject the null-hypothesis because the chance of having a
  sample with difference of 0.33 (in magnitude) is less than
  0.02.
 If the null hypothesis is true (that the two perceptions are
  being the same, i.e. the difference is 0) the value of 0.33 will
  have 95% chances greater than the sampling mean of the
  population (with true population difference equal to 0)
  between 0.12 and 0.54.
 Testing the differences between two samples




 Research question: Does male and female have a different
  level of trust towards social workers?
 Null hypothesis: Male and female have the same mean
  value of trust towards social workers.
 Alternative hypothesis: Male and female have a different
  mean value of trust towards social workers.
                                         Group Statistics



        性別              N                 Mean              Std. Deviation         Std. Error Mean

社工    男                     230                6.30                  1.825                        .120
      女                     273                6.27                  1.883                        .114



                                                             Independent Samples Test
                               Levene's Test for Equality
                                     of Variances                                               t-test for Equality of Means
                                                                                                                                      95% Confidence Interval
                                                                                                                        Std. Error       of the Difference
                                     F          Sig.            t            df         Sig. (2-tailed) Mean Difference Difference      Lower        Upper
社工   Equal variances assumed         .836         .361           .152             501           .879             .025          .166        -.301       .352
     Equal variances not assumed                                 .153    491.308                .879             .025          .166        -.300       .351

				
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