191
5.5.5
Chi-Square Test
This is the statistical tool used to carry out tests for several proportions (Agbadudu, 1994). If chi-square is denoted by X2, i.e. squaring
the difference between the observed and the expected frequencies and then divide the result by the expected frequencies. This test is
closely approximated by chi-square distribution, and it has a degree of freedom k-1. If X2 = 0, then there is a perfect agreement between
observed and expected frequencies. The greater the discrepancy between the observed and the expected frequencies, the larger the value
of X2. Hence, we reject Ho for Ha if the calculated X2 is greater than X21-n …kj (using the table).
Conditions to be satisfied in Chi-Square Test: (i). Each observation or frequency must be independent of all other observations (ii). The
sample size must be reasonably large in order that the difference between the actual and expected observations be normally distributed
(iii). No expected frequencies or observations should be small.
Frequencies
Table 5.6: Mix 1:2:4 Unwashed Specimen A and Mix 1:2:4 Washed Specimen A
cubeweightU
Observed N Expected N
7.00
1
1.3
7.20
1
1.3
7.30
2
1.3
Total
4
cubeweightW
Observed N Expected N
7.00
1
1.3
7.90
1
1.3
8.10
2
1.3
Total
4
crushingvalU
Observed N Expected N
.40
1
1.0
60.50
1
1.0
125.00
1
1.0
138.90
1
1.0
Total
4
192
crushingvalW
compresstrengthU
Observed N Expected N
compresstrengthW
Observed N Expected N
143.00
1
1.0
188.90
1
1.0
2.70
1
1.0
227.30
1
1.0
3.40
1
1.0
6.40
1
1.0
316.00
1
1.0
5.60
1
1.0
8.40
1
1.0
Total
4
6.20
1
1.0
10.10
1
1.0
Total
4
14.10
1
1.0
Total
4
Observed N Expected N
Test Statistics
crushingval crushingval compresstret compresstre
cubeweightU cubeweightW
U
W
U
tW
Chi-Square
0.500
0.500
0.000
0.000
0.000
0.000
2
2
3
3
3
3
0.779
0.779
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Lower
Bound
0.905
0.905
0.905
0.905
0.905
0.905
Upper Bound
1.000
1.000
1.000
1.000
1.000
1.000
df
Asymp. Sig.
Monte Carlo
Sig.
Sig.
95% Confidence
Interval
Results obtained from Table 5.6 above indicated that the observed and expected frequencies are equal to zero for cube weight test but are
not equal to zero at 95% confidence interval for crushing values and compressive strengths respectively. The null hypothesis is therefore
upheld.
193
Table 5.7: Mix 1:2:4 Unwashed specimen C and Mix 1:2:4 Washed Specimen A
cubeweightU
cubeweightW
Observed N Expected N
Observed N Expected N
crushingvalU
7.70
1
1.3
7.80
1
1.3
7.00
1
1.3
8.00
2
1.3
7.90
1
1.3
70.90
1
1.0
Total
4
8.10
2
1.3
128.30
1
1.0
Total
4
188.90
1
1.0
223.10
1
1.0
Total
4
Observed N Expected N
crushingvalW
compresstretU
Observed N Expected N
compresstretW
Observed N Expected N
143.00
1
1.0
188.90
1
1.0
3.20
1
1.0
227.30
1
1.0
70
1
1.0
6.40
1
1.0
316.00
1
1.0
8.40
1
1.0
8.40
1
1.0
Total
4
9.80
1
1.0
10.10
1
1.0
Total
4
14.10
1
1.0
Total
4
Observed N Expected N
194
Test Statistics
cubeweight cubeweight crushingval crushingval compresstret compresstret
U
W
U
W
U
W
Chi-Square
0.500
0.500
0.000
0.000
0.000
0.000
2
2
3
3
3
3
0.779
0.779
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Lower
Bound
0.905
0.905
0.905
0.905
0.905
0.905
Upper
Bound
1.000
1.000
1.000
1.000
1.000
1.000
df
Asymp. Sig.
Monte Carlo
Sig.
Sig.
95% Confidence
Interval
Table 5.7 results shows that, the observed and expected frequencies are equal to zero for cube weight test but are not equal to zero at
95% confidence interval for crushing values and compressive strengths respectively. The null hypothesis is therefore upheld.
Table 5.8: Mix 1:3:6 Unwashed Specimen F and Mix 1:3:6 Washed Specimen B
cubeweightU
cubeweightW
Observed N Expected N
crushingvalW
Observed N Expected N
5.90
1
1.3
6.00
1
1.3
7.00
2
2.0
6.20
2
1.3
7.10
2
2.0
Total
4
Total
4
Observed N Expected N
57.40
2
1.3
68.80
1
1.3
85.60
1
1.3
100.00
1
1.3
Total
5
195
crushingvalU
compresstretU
Observed N Expected N
compresstretW
Observed N Expected N
96.30
1
1.0
141.50
1
1.0
2.60
1
1.0
166.80
1
1.0
3.10
1
1.0
4.30
1
1.0
253.80
1
1.0
3.80
1
1.0
6.30
1
1.0
Total
4
4.40
1
1.0
7.40
1
1.0
Total
4
11.30
1
1.0
Total
4
Observed N Expected N
Test Statistics
cubeweight crushingval crushingval compresstret compresstret
cubeweightU
W
U
W
U
W
.500a
.000b
.600c
.000d
.000d
.000d
2
1
3
3
3
3
.779
1.000
.896
1.000
1.000
1.000
1.000e
1.000e
1.000e
1.000e
1.000e
1.000e
Lower
Bound
.905
.905
.905
.905
.905
.905
Upper
Bound
1.000
1.000
1.000
1.000
1.000
1.000
Chi-Square
df
Asymp. Sig.
Monte Carlo
Sig.
Sig.
95% Confidence
Interval
196
From Table 5.8, it was observed that the observed and expected frequencies are equal to zero for cube weight test but are not equal to
zero at 95% confidence interval for crushing values and compressive strengths respectively. The null hypothesis is therefore upheld.
In summary, since all the sampled test results indicated that the null hypothesis was upheld, this implied that the postulated
hypothesis that the null hypothesis design calculations and mix ratio specification parameters normally used for the production and
management of site in-situ concrete is inadequate is hereby upheld.
PAIRED T – TEST
5.5.6
Decision Rule for Paired Specimen Tests Statistics: In the paired sample test method of analysis, the parameters tested are the means,
standard deviation, correlation and confidence interval between one and the other at 95% significance for two-tailed test. If there is
remarkable difference between the specimens being paired then the null hypothesis is upheld.
Table 5.9: Mix 1:2:4 Unwashed Specimen A and Mix 1:2:4 Washed Specimen A
Paired Specimens Statistics
Mean
Std.
Deviation
N
Std. Error
Mean
Pair 1 cubeweightU
7.7750
4
.52520
.26260
cubeweightW
7.2000
4
.14142
.07071
Pair 2 crushingvalU
100.1000
4
37.92071
18.96035
crushingvalW
218.8000
4
73.39332
36.69666
Pair 3 compresstretU
4.4750
4
1.68795
.84397
compresstretW
9.7500
4
3.27058
1.63529
197
Paired Specimens Correlations
N
Correlation
Sig.
Pair 1 cubeweightU &
cubeweightW
4
-.090
.910
Pair 2 crushingvalU &
crushingvalW
4
.925
.075
Pair 3 compresstretU &
compresstretW
4
.924
.076
Paired Specimens Test
Paired Differences
95% Confidence Interval of
the Difference
Mean
Std.
Deviation
Std. Error
Mean
Lower
Upper
t
Sig. (2tailed)
df
Pair 1 cubeweightU cubeweightW
.57500
.55603
.27801
-.30976
1.45976
2.068
3
.130
Pair 2 crushingvalU crushingvalW
-118.70000
40.95689
20.47844
-183.87155
-53.52845
-5.796
3
.010
-5.27500
1.83007
.91504
-8.18705
-2.36295
-5.765
3
.010
Pair 3 compresstretU compresstretW
198
The lowest mean pair 3 for Unwashed specimen (4.4750) Table 5.9 as against the highest mean pair 2 for Washed specimens was
(218.8000), a difference of (214.325). The least standard deviation value pair 1 cube weight for Washed specimens (0.14142) whereas
pair 2 crushing value for Washed specimens had (73.39332). Also, the test analysis indicates that there is 0.090 correlation coefficient
for pair 1 as against the highest correlation coefficient of 0.925 for pair 3. The implication of the above statistical test results is that,
there is a remarkable difference between the mean, standard deviation and correlation coefficient between the specimens of Washed and
Unwashed materials. Using the mean and standard deviation test parameters, we can then conclude that the Washed specimens will
produce better quality concrete than the Unwashed specimens.
T able 5.10: Mix 1:2:4 Unwashed Specimen E and Mix 1:2:4 Washed specimen A
Paired Specimens Statistics
Mean
Std.
Deviation
N
Std. Error
Mean
Pair 1 cubeweightU
6.2250
4
.51881
.25941
cubeweightW
7.7750
4
.52520
.26260
Pair 2 crushingvalU
82.7250
4
24.50284
12.25142
crushingvalW
218.8000
4
73.39332
36.69666
Pair 3 compresstretU
3.7000
4
1.06771
.53385
compresstretW
9.7500
4
3.27058
1.63529
199
Paired Specimens Correlations
N
Correlation
Sig.
Pair 1 cubeweightU &
cubeweightW
4
-.963
.037
Pair 2 crushingvalU &
crushingvalW
4
.981
.019
Pair 3 compresstretU &
compresstretW
4
.982
.018
Paired Specimens Test
Paired Differences
95% Confidence Interval of
the Difference
Mean
Std.
Deviation
Std. Error
Mean
Lower
Upper
t
Sig. (2tailed)
df
Pair 1 cubeweightU cubeweightW
-1.55000
1.03441
.51720
-3.19597
.09597
-2.997
3
.058
Pair 2 crushingvalU crushingvalW
-136.07500
49.57677
24.78838
-214.96270
-57.18730
-5.489
3
.012
-6.05000
2.23084
1.11542
-9.59977
-2.50023
-5.424
3
.012
Pair 3 compresstretU compresstretW
From Table 5.10, the lowest mean pair 3 for Unwashed specimen (3.7000) as against the highest mean pair 2 for Washed specimens was
(218.8000), a difference of (215.100). The least standard deviation value pair 1 cube weight for Washed specimens (0.51881) whereas
200
pair 2 crushing value for Washed specimens had (73.39332). Also, the significant 2-tailed test analysis indicates that both pairs 2 & 3
are at the same level (0.012) whereas pair 1 recorded 0.058 significant 2-tailed value. The implication of the above statistical test results
is that, there is a remarkable difference between the mean, standard deviation and correlation coefficient between the specimens of
Washed and Unwashed materials. We can then conclude that the Washed specimens will produce better quality concrete than the
Unwashed specimens.
Table 5.11: Mix 1:3:6 Unwashed Specimen F and Mix 1:3:6 Washed Specimen B
Paired Specimens Statistics
Mean
Std.
Deviation
N
Std. Error
Mean
Pair 1 cubeweightU
6.0750
4
.15000
.07500
cubeweightW
7.0500
4
.05774
.02887
Pair 2 crushingvalU
77.9500
4
18.71497
9.35748
crushingvalW
164.6000
4
66.23187
33.11593
Pair 3 compresstretU
3.4750
4
.78899
.39449
compresstretW
7.3250
4
2.94434
1.47217
Paired Specimens Correlations
N
Correlation
Sig.
Pair 1 cubeweightU &
cubeweightW
4
-.192
.808
Pair 2 crushingvalU &
crushingvalW
4
.968
.032
Pair 3 compresstretU &
compresstretW
4
.966
.034
201
Paired Specimens Test
Paired Differences
95% Confidence Interval
of the Difference
Std.
Std. Error
Mean Deviation
Mean
Pair 1 cubeweightU – -.97500
cubeweightW
Lower
Upper
.17078
.08539
-1.24675
Pair 2 crushingvalU –
- 48.35111
crushingvalW 86.650
00
24.17556
-163.58741
-9.71259
1.09583
-7.33741
-.36259
Pair 3 compresstretU
3.8500
compresstretW
0
2.19165
t
-.70325 -11.418
df
Sig. (2-tailed)
3
.001
-3.584
3
.037
-3.513
3
.039
From Table 5.11, the lowest mean pair 3 for Unwashed specimen recorded was (3.4750) as against the highest mean pair 2 for Washed
specimens of value (164.6000), a difference of (161.125). The least standard deviation value pair 1 cube weight for Washed specimens
(0.05774) whereas pair 2 crushing value for Washed samples had (66.23187). Also, the test analysis indicates that there is 0.192
correlation coefficient for pair 1 as against the highest correlation coefficient of 0.968 for pair 2. Also using the significant 2-tailed
parameter, all the pairs are less than 0.005 significant value. The implication of the above statistical test results is that, there is a
remarkable difference between the mean, standard deviation and correlation coefficient between the specimens of Washed and
Unwashed materials. Using the mean and standard deviation test parameters, we can then conclude that the Washed specimens will
produce better quality concrete than the Unwashed specimens. It can then be concluded that the null hypothesis design calculations and
mix ratio specification parameters normally used for the production and management of site in-situ concrete is inadequate is hereby
upheld. Meaning that, additional parameters for the production of good in-situ concrete had become inevitable in Nigeria and had been
advocated in this study.