# Hypothesis testing

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```					Hypothesis testing

Research II Class 5
The logic of hypothesis testing
 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
   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

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

One-Sam ple Test
Test Value = 6
95% Confidence Interval
of the Difference
t           df         Sig. (2-tailed) Mean Difference   Lower        Upper

 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.
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

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

Equal variances not assumed                                 .153    491.308                .879             .025          .166        -.300       .351

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