EXPERIMENT 1
Using the Analytical Balance and Piso1 Statistics
I.
Data and Results
Table 1. Masses of coins obtained by the two weighing methods (before washing with acetone).
Year Category
Sample No.
Mass (g)
Weighing Individually
Weighing by Difference
2000
5.8907
5.8909
2001
6.0051
6.0044
2002
6.0153
6.0154
2004
2003
6.0107
6.0108
2004
5.3358
5.3355
2009
5.3576
5.3572
2010
5.3380
5.3388
2011
5.4318
5.4310
2004
2012
5.3654
5.3651
2013
5.2998
5.3002
Table 2. Masses of coins obtained by the two weighing methods (after washing with acetone).
Year Category
Sample No.
Mass (g)
Weighing Individually
Weighing by Difference
2000
5.8904
5.8905
2001
6.0049
6.0047
2002
6.0154
6.0152
2004
2003
6.0108
6.0106
2004
5.3357
5.3355
2009
5.3573
5.3573
2010
5.3377
5.3380
2011
5.4319
5.4318
2004
2012
5.3650
5.3650
2013
5.3000
5.3002
Table 3. Statistical parameters for masses of coins in each year category obtained by weighing by difference (before washing with
acetone).
Year
STATISTICS
Categor
Mean
S
RSD
S.E.M.
Range
Rel. Range
95% C.I.
Spooled
y
2004
5.8514
0.2930
5.0073
0.1310
0.6799
11.6194
(5.4871,6.2157)
0.3088
2004
5.3585
0.0477
0.8902
0.0213
0.1308
2.4410
(5.2992,5.4178)
0.0520
Data
Year
Category
Suspected
Values
H: 6.0154
Table 4. Grubbs test for outliers
GCalc
Gcrit
Conclusion
0.5597
1.672
Gcalculate<Gtable, Sample Accepted
L: 5.3355
1.7608
1.672
Gcalculate>Gtable, Sample Rejected
2004
H: 5.4318
L: 5.3002
1.5199
1.2222
1.672
1.672
Gcalculate<Gtable, Sample Accepted
Gcalculate<Gtable, Sample Accepted
2004
H: 6.1702
L: 5.2467
1.1349
2.1610
2.8675
2.8675
Gcalculate<Gtable, Sample Accepted
Gcalculate<Gtable, Sample Accepted
H: 5.4884
2.0257
2.8675
Gcalculate<Gtable, Sample Accepted
L: 5.2863
1.9471
2.8675
Gcalculate<Gtable, Sample Accepted
2004
Group
Pooled
2004
Table 5. Recalculated statistical parameters for masses of coins in each year category obtained by weighing by difference.
Year
STATISTICS
Categor
Mean
S
RSD
S.E.M.
Range
Rel. Range
95% C.I.
Spooled
y
2004
5.9804
0.0597
0.9983
0.0210
0.1245
2.0818
(6.0627,5.9084)
0.0708
2004
5.3585
0.0477
0.8902
0.0213
0.1308
2.4410
(5.2992,5.4177)
0.0514
Table 6. Comparison of masses of coins from two different year categories. (t-test)
Method
tcalc
tcrit
Conclusion
Weighing individually
Weighing by difference
83.5336
319.68
Reject null hypothesis
Reject null hypothesis
Table 7. Comparison of two weighing methods
Calculated
Critical
2.365
0.3804
Test
Paired-t test
F test
II.
2.78
2.78
1.0046
Conclusion
Accept null hypothesis
5.05
Accept null hypothesis
Calculations
Table 3. Statistical parameters for masses of coins in each year category obtained by weighing by difference.
2004
MEAN
x =
STANDARD
DEVIATION
s=
RELATIVE
STANDARD
DEVIATION
STANDARD
ERROR OF
THE MEAN
RANGE
RELATIVE
RANGE
xi = 5.8909+6.0044+ 6.0154+6.0108+5.3355
(x i x )2
n1
0.3435
51
s
0.2930
RSD= x 100
= 5.8514
x
SEM =
s
n
0.2930
5
= 5.8514
= 0.2930
x 100 =
5.0073
= 0.1310
range=x largest x smallest = 6.0154 - 5.3355 = 0.6799
Relative range=
range
x 100
x
0.6799
= 5.8514
x 100 = 11.6194
@ 95%
CONFIDENCE
INTERVAL
Confidence interval=x
t n1 s
n
= 5.8514
( 2.78 ) ( 0.2930 )
5
(5.4871,6.2157)
Spooled
S pooled =
( n11 ) S 21+ ( n21 ) S22 +(nk 1)S 2k
N 1+N +N
2
4 ( 0.85856605 )+ 4 ( 0.71510105 ) + 4 ( 0 .086186388 )+ 4 ( 0.1069423 )+ 4 ( 0.090103235 )
+4 ( 0.118235037 ) + 4 ( 0.087752573 ) +4 (0.001026503)
S p=
5+ 5+5+5+5+5+5+ 58
S p = 0.3088
2004
MEAN
x =
xi
5.3572+5.3388+5.431+5.3651+5.3002
= 5.3585
5
( x i x )2
0 . 0091
51
STANDARD
DEVIATION
s=
RELATIVE
STANDARD
DEVIATION
s
0.0477
RSD= x 100
= 5.3585
x
STANDARD
ERROR OF
THE MEAN
SEM =
RANGE
RELATIVE
RANGE
n1
s 0.0477
=
n
5
= 0.0477
x 100
= 0.8902
= 0.0213
range=x largest x smallest = 5.4310 - 5.3002 = 0.1308
Relative range=
range
x 100
x
0.1308
= 5.3585 x 100 = 2.4410
@ 95%
CONFIDENCE
INTERVAL
Confidence interval=x
t n1 s
= 5.3585
( 2.78 ) ( 0.0477 )
5
=(5.2992,5.4178)
Spooled
S pooled =
( n11 ) S 21+ ( n21 ) S22 +(nk 1)S 2k
N 1+N +N
2
SP =
4 ( 0.002272118 ) + 4 ( 0.001176157 )+ 4 ( 0.008175448 ) + 4 ( 0.0033488 ) +4 ( 0.002468887 )
+4 ( 0.000891242 )+ 4 ( 0.002313743 ) + 4(0.001026503)
5+ 5+5+5+5+5+5+ 58
S p = 0.0520
Table 4. Grubbs test for outliers.
Group
2004
|questionables valuex| |6.01545.8514
|
0.2930
H :G calc=
Gcrit= 1.672
0.5597
|questionables valuex| |5.33555.8514
|
0.2930
L:Gcalc =
1.7608
2004
H:
|questionables value x |= |5.43105.3585
|
0.0477
Gcalc =
1.5199
L:
.3585
|questionables value x |=|5.30025
|
0.0477
Gcalc =
Pooled
2004
1.2222
Gcrit= 1.672
|questionables valuex|=|6.17025.8522
|=1.1349
0.2802
H :G calc=
Gcrit = 1.672
|questionables valuex|=|5.24675.8522
|=2.1610
0.2802
L:Gcalc =
|questionables valuex|=|5.48845.3856
|=2.0157
0.0510
H :G calc=
Gcrit = 1.672
|questionables valuex|=|5.28635.3856
|=1.9471
0.0510
L:Gcalc =
Table 5. Recalculated statistical parameters for masses of coins in each year category obtained by weighing by differences.
2004
MEAN
x =
xi = 5.8909+6.0044+ 6.0154+6.0108
(x i x )2
STANDARD
DEVIATION
s=
RELATIVE
STANDARD
DEVIATION
s
RSD= x 100
x
STANDARD
ERROR OF
THE MEAN
SEM =
RANGE
RELATIVE
RANGE
@ 95%
CONFIDENC
E INTERVAL
= 5.9804
n1
0.0107
41
0.0597
= 5.9804
s 0.0597
=
n
4
= 0.0597
x 100 = 0.9983
= 0.0210
range=x largest x smallest = 6.0154 - 5.8909 = 0.1245
Relative range=
range
x 100
x
Confidence interval=x
=(5.8974,6.0634)
t n1 s
0.1245
= 5.9804 x 100 = 2.0818
= 5.9804
( 2.78 ) (0.0597)
4
Spooled
S pooled =
( n11 ) S 21+ ( n21 ) S22 +(nk 1)S 2k
N 1+N +N
2
Sp=
0.011439036
3 ( 0.003578469 ) +3 ( 0.000304407 ) +3 ( ) +3 ( 0.004157667 ) +3 ( 0.002698143 )
+3 ( 0.011975033 ) +3 ( 0.00397247 ) +3(0.00192158)
3+3+3+ 3+3+3+3+38
Sp = 0.0708
2004
MEAN
STANDARD
DEVIATION
RELATIVE
STANDARD
DEVIATION
STANDARD
ERROR OF
THE MEAN
x =
s=
xi
5.3572+5.3388+5.431+5.3651+5.3002
= 5.3585
5
( x i x )2
n1
0.0091
51
s
0.0477
RSD= x 100
= 5.3585
x
SEM =
s 0.0477
=
n
5
= 0.0477
x 100
0.8902
0.0213
RANGE
range=x largest x smallest = 5.4310 - 5.3002 = 0.1308
RELATIVE
RANGE
@ 95%
CONFIDENC
E INTERVAL
Relative range=
range
x 100
x
Confidence interval=x
t n1 s
0.1308
x 100
5.3585
= 5.3585
= 2.4410
( 2.78 ) (0.0477)
4
=(5.2992,5.4177)
S pooled =
Spooled
( n11 ) S 21+ ( n21 ) S22 +(nk 1)S 2k
N 1+N +N
2
4 ( 0.002272118 ) +3 ( 0.000470817 ) +4 ( 0.008175448 )+ 4 ( 0.0033488 ) + 4 ( 0.002468887 )
+4 ( 0.000891242 ) +3 ( 0.001719817 ) + 4(0.001026503)
S p=
5+ 4+5+5+ 5+5+4 +58
S p = 0.0514
Table 6. Comparison of masses of coins from two different year categories (t-test).
Ho: 1 = 2, or 1 - 2 = 0 ; otherwise reject the Ho.
Weighing Individually
s=
s=
2
1
Weighing by Difference
( n1 1 ) s + ( n21 ) s 2
s=
(n 1+ n22)
( 3 x 0.0257 )+(4 x 0.0002)
( 4 +52 )
s=
( n1 1 ) s 12 + ( n21 ) s 22
( 3 x 0.0036 )+(4 x 0.0023)
( 4 +52 )
s=0.0111
s=0. 0029
Degrees of freedom: 4+52=7
Degrees of freedom: 4+52=7
TCalc
x 1x 2
t=
s
1 1
+
n1 n2
TCalc
5.9805 5.3585
(n 1+ n22)
0.0111
1 1
+
4 5
x 1x 2
t=
s
1 1
+
n1 n2
5.9804 5.3585
1 1
0.0029
+
4 5
319.68
83.5336
To solve the variance for t-test of two weighing methods, use the Table 1A-B, 1F-G in theAppendix. . Note:
coins minted before 2004 and
s 22
= coins minted after 2004.
Paired t-test
F test
sd
di
d=
n
sd =
0.0006
d=
9
sd =
d=0.0000667
s 12
(d id )2
0.00000221
91
n1
s d =0.000526
2
1
s 1=
(x 1x )2
s 1=
0.8806
91
n1
s 22
s 2=
( x 2x )2
s 2=
0.8769
91
n1
s 1=0.3318
s 2=0.3311
s 21=0 .1101
s 22=0.1096
Table 7. Comparison of two weighing methods.
2
To solve the values of
individually and
s 22
s1
s2
: use the Table 1C, 1H, 1M. Note:
s1
= the variance of weighing
= the variance of weighing by difference.
Paired T test and F test
Paired t-test
CALCULATED
t=
t=
d n
sd
0.0000667 ( 9 )
0.000526
t=0.3804
F test
CRITICAL
2.365
CALCULATED
CRITICAL
F=
F=
s1
2
s2
0.1101
0.1096
F= 1.0046
5.05
