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Chapter 2
Two samples
2
...
g
...
IQ’s of babies bottle fed or breast fed
...
Saliva concentration of respondents who were lying versus those who were
truthful
...
Time to complete task when instructed in two different ways
...
Weight gain of anorexic girls when subjected to two different treatments
...
There would be
some ethical issues associated with conducting a strict experiment whereby the
treatments were randomly allocated to the babies
...
Taking example 4, suppose a random sample of 16 girls is taken all of
whom are anorexic
...
To assess the effectiveness of the treatments it is usual
that one of the treatments is a ‘control’ i
...
, no treatment or the usual treatment
...
Parametric v non parametric
A parametric test is based on several assumptions about the parameters of the
populations from which the samples were taken
...
It is relatively easy (usually) to test the assumptions underpinning a parametric test
In non-parametric tests no assumptions are made about the underlying distribution
and the tests can be applied to nominal or ordinal variables
...
Both point and interval estimation are important
...
Recapping on the standard
procedure:
4
General Structure of a Hypothesis Test
1
...
The hypotheses are
statements about the population parameters
...
State the test statistic
3
...
Define the level of significance of the test (the probability of rejecting when it is
true) and hence the critical region
...
Calculate the observed value of the test statistic using the sample data
...
Reject if observed value of test statistic falls in critical region or p-value is less
than
...
7, State the conclusions clearly in non-technical terms
...
2
...
The two
treatments are randomly assigned to the experimental material and the appropriate
measurements taken
...
Two groups of healthy adults are randomly assigned to one of two diets: Normal
or High Fibre
...
1
High Fibre
12
...
0
13
...
6
13
...
7
20
...
0
22
...
8
23
...
8
Assumptions:
The two measures of vitamin D are Normally distributed
...
Notation
2
For the Normal diet, let be the sample mean, s1 be the sample
variance
For the High Fibre, let be the sample mean, be the sample variance
For the Normal diet, let be the population mean, be the population
variance
For the High Fibre, let be the population mean, be the population
5
variance
Hypotheses:
v
T
Test Statistic:
x1 x 2
1
1
sp
n
1 n2
s2 p
n1 1s1 2 n2 1s 2 2
n1 n2 2
where
If is true then T ~
...
e
...
20 or T < -2
...
70 =18
...
70 18
...
09
=
= 2
...
06
1 1
30
...
97 > 2
...
Conclude that there is a significant difference between the vitamin D levels of the two
groups
...
09 in the high fibre diet group
...
3
Two samples: matched pairs
Here the two samples are not independent
...
Pairing is sometimes achieved by using twins, using the same
subject twice, or simply pairing by some other characteristic
...
The results
were:
Subject
Post Exercise
Control
1
13
...
5
2
14
...
6
3
42
...
6
4
20
...
4
5
19
...
7
6
17
...
6
Consider the sample of six differences and assume that the population of differences
is Normally distributed
...
1
2
...
2
0
...
N is the sample size
...
5
3
...
57 or T < - = -2
...
Hence T is not greater than 2
...
57
...
Conclude that on average there is not a significant difference in the growth hormone
level when exercise is taken compared to that when there is no exrecise
...
Here all the subjects showed an increase in hormone level on days when they had
taken exercise
...
This would suggest that hormone
levels tended to be increased after exercise
...
The average difference between exercise and non-exercise is not significant,
using a two-sided 5% test
...
Large amount of variability
...
‘rigid’ interpretation of significance levels
...
Comparison of 2-independent samples v matched pairs
...
It will
be more efficient if there is a large amount of variability between blocks
...
4
...
A further
discussion of the efficiency of experimental design is in Chapter 4
...
4 SPSS output
2
...
1 Two independent samples:
Assuming the data are in two separate columns, one for the variable to be tested and
one for the grouping variable, choose
Analyze
Compare Means
Independent-Samples T Test
This opens the Independent-samples T Test dialog box
...
Click on the variable to be tested, then click > to move it to the Test Variable(s)
box
...
Click on the Grouping variable, then click on > to move it to the Grouping Variable
box
...
Click on Define Groups to open the Define Groups dialog box
...
In the Group 1 box, enter the value used to represent group 1
...
5
...
Click OK
...
The Levene’s test evaluates whether the population variances for the two groups are
equal
...
39 and p = 0
...
968, p =
0
...
The test indicates that there is a significant difference because the p-value is
less than 0
...
T-Test
Group Statistics
Vitamin D
Type of diet
Normal diet
High fibre
N
Std
...
Error
Mean
Deviation
Mean
27
...
444
2
...
614
5
...
098
6
7
Independent Samples Test
Levene's Test for
Equality of Variances
Vitamin D
Equal variances
assumed
Equal variances not
assumed
F
Sig
...
390
t-test for Equality of Means
t
...
(2-tailed)
df
Mean
Difference
Std
...
968
11
...
086
3
...
349
15
...
973
10
...
013
9
...
056
2
...
830
2
...
1 Paired t-tests
Assuming that the data are in two columns, choose
Analyze
Compare Means
Paired-Samples T Test
to open the Paired-Samples T Test dialog box
...
Click on the first variable in the pair
...
Click on the second variable
...
3
...
4
...
The output is:
8
T-Test
Paired Samples Statistics
Pair
1
Mean
Growth Hormone Level
Post Exercise
Growth Hormone Level
without exercise
Std
...
Error
Mean
21
...
837
4
...
067
6
4
...
940
Paired Samples Correlations
Pair
1
N
Growth Hormone Level
Post Exercise & Growth
Hormone Level without
exercise
Correlation
6
Sig
...
809
...
200
Std
...
530
Std
...
074
95% Confidence
Interval of the
Difference
Lower
Upper
-1
...
102
Here there is no evidence to reject the null hypothesis since p=0
...
05
...
(2-tailed)
df
2
...
100
Practical 2: Two Samples Test
Logging on:
Choose the following options by positioning the pointer and Click (or Double Click)
Start
Programs
Statistics
SPSS for Windows
SPSS 14 for Windows
This opens the SPSS for Windows dialog box
...
Click OK
...
The problem: Fifteen retarded individuals were instructed in self-care skills
throughout a six-month period using imitation as the means of teaching the skills
...
After each six-month period the individuals were rated on the level of assistance to
carry out the tasks
...
The data were:
Ratings
Subject
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Imitation
14
11
19
8
4
9
12
5
14
17
18
0
2
8
6
Physical Guidance
10
13
15
5
3
6
7
9
16
10
13
1
2
3
6
1
...
2
...
3
...
The variable now appears as Variable 1 in
Current Selections
4
...
It now appears as Variable 2 in Current Selections
...
Click on > to move the two variables to the Paired Variables box
...
Click OK
...
Print the VIEWER window (you may store it if you wish as a TEXT file to be
incorporated into a WORD document
...
The problem: In an experiment, students were used and they were given caffeine
before performing a task
...
The results were for
the task were:
Expectation
Beneficial
19
Detrimental 14
15
18
22
17
13
12
18
21
15
21
20
24
25
14
22
(one student had to go home early!)
3
...
Create a grouping variable to distinguish between the two treatment groups,
assigning the value 1 for Beneficial Effect and the value 2 for Detrimental Effect
...
Click Analyze
Compare Means
Independent-Samples T Test
to open the Independent-Samples T Test dialog box
...
Click on the variable to be tested, then click > to move it to the Test Variable(s)
box
...
Click on the Grouping variable, then click on > to move it to the Grouping
Variable box
...
Click on Define Groups to open the Define Groups dialog box
...
In the Group 1 box, enter the value used to represent group 1
...
10
...
Click OK
...
Print VIEWER window as before
...
2
...
4
...
Analyse the data, testing whether there are any significant differences
between the treatments
...
Summarise your conclusions