Transcript c. The t Test with Multiple Samples

c.
The t Test with Multiple Samples
Till now we have considered replicate
measurements of the same sample. When
multiple samples are present, an average
difference is calculated and individual
deviation from a mean difference is
calculated and used to calculate a difference
standard deviation, Sd which is used in a
successive step to calculate t.
+ t = DN1/2/Sd
Sd = [S ( Di – D )2 / (N-1)]1/2
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Sd is the standard deviation of the difference, Di is
the difference between a result obtained by the
standard method subtracted from that obtained
by the proposed method for the same sample. D
is the average of all differences.
Example
Mercury in multiple samples was determined
using a standard method and a new suggested
method. Six different samples were analyzed
using the two procedures giving the following
results in ppm:
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Sample No.
1.
2.
3.
4.
5.
6.
New Method
10.3
12.7
8.6
17.5
11.2
11.5
Standard method
10.5
11.9
8.7
16.9
10.9
11.1
Find the standard deviation of the difference. If the two
methods have comparable precisions, find whether
there is any significant difference between the
results of the two methods at the 95% confidence
level. The tabulated t value for five degrees of
freedom at 95% confidence level is 2.571.
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Sample No. New Method Standard method
Di
1.
10.3
10.5
-0.2
2.
12.7
11.9
+0.8
3.
8.6
8.7
-0.1
4.
17.5
16.9
+0.6
5.
11.2
10.9
+0.3
6.
11.5
11.1
+0.4
_______________________________________________
SDi = 1.8
D = 1.80/6 = 0.30
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S ( Di – D )2 = { (-0.2-0.3)2 + (+0.8-0.3)2 + (-0.1-0.3)2 +
(+0.6-0.3)2 + (+0.3-0.3)2 + (+0.4-0.3)2 } =
{0.25+0.25+0.16+0.09+0+0.01}
S ( Di – D )2 = 0.76
Sd = ( S( Di – D )2 / (N-1) )1/2
Sd =
(0.76/5)1/2
= 0.39
+ t = DN1/2/sd
+ t = 0.30x61/2/0.39 =1.88
The calculated t value is less than the tabulated t value
which means that there is no significant difference
between the results of the two methods.
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The Q Test
In several occasions, when replicate experiments are
done one of the data point may look odd or faulty.
The analyst is confused whether to keep it or reject
it. The Q test provides a means to judge if it should
be retained or rejected. This can be done by applying
the Q test equation:
Q = a/w
Where a is the difference between the suspected result
and the result nearest to it in value, w is the
difference between highest and lowest results.
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Once again, if the calculated Q value is less
than the tabulated value, then the
suspected data point should be retained.
In contrast to F and t tests the statistical
value of Q depends on the number of data
points rather than the number of degrees
of freedom.
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Example
In the replicate determination of gold you got the
following results: 96, 99, 97, 94, 100, 95, and
72%. Check whether any point should be
excluded at the 95% confidence level. Tabulated
Q95% = 0.568 for 7 observations
Arrange results: 72, 94, 95, 96, 97, 99, 100
Q = a/w
Qcalc = (94-72)/(100 - 72) = 0.79
Qcalc > Qtab
The point 72% should be rejected.
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Example
In the replicate determination of gold you got the
following results: 96, 99, 97, 94, 100, 95, and 88%.
Check whether any point should be excluded at the
95% confidence level. Tabulated Q95% = 0.568 for 7
observations.
Arrange results: 88, 94, 95, 96, 97, 99, 100
Solution
Q = a/w
Qcalc = (94-88)/(100-88) = 0.50
Qcalc < Qtab
The point 88% should be retained.
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Linear Least Squares
Frequently, an analyst constructs a calibration
curve using several standards and draws a
straight line among the data points in the graph.
In many cases, the line does not cross all points
and the analyst starts judging where the straight
line should pass. Human judgment is not perfect
and, unfortunately, may be biased. The method
of linear least squares is a mathematical method
that help us choose the best path of the straight
line.
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A least-squares plot gives the best straight line through
experimental points. Excel will do this for you.
Residual = yi – yl
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It is well known that the equation of a straight line is
mathematically represented by
y = mx + b
Where m is the line and b is the line intercept, x and y are
variables.
The slope, m, can be calculated from the relationship
m = {Sxiyi – [(SxiSyi)/n]}/{ Sxi2 – [(Sxi)2/n]}
b = y – mx
x, y are average values of xi and yi.
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The standard deviation of any of the yi points
(Sy) is given by the relation
Sy = {([Syi2 – (( Syi)2/n)] – m2 [Sxi2 – ((
Sxi)2/n)])/(n-2)}1/2
The uncertainty in slope can then be
calculated from Sy as follows
Sm = {Sy2/ [Sxi2 – (( Sxi)2/n)]}1/2
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Example
Using the following data and without plotting,
if the fluorescence of a riboflavin sample
was 15.4 find its concentration.
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(Sxi)2 = 2.250
x = (Sxi)/n = 1.500/5 = 0.300
y = (Syi)/n = 83.6/5 = 16.72
m = {Sxiyi – [(SxiSyi)/n]}/{ Sxi2 – [(Sxi)2/n]}
Substitution in the equation above gives
m = {46.6 – [(1.500*83.6)/5]}/ {0.850 –[(2.250/5]}
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m = 53.75
This Excel plot gives the same results for slope and intercept
as calculated in the example.
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To calculate b we use the equation
b = y – mx
b = 16.72 – 53.75*0.300 = 0.60
Now we are ready to calculate the sample
concentration
y = mx + b
15.4 = 53.75*x + 0.60
x = 0.275 ng/L
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Correlation Coefficient (r)
When the points that are supposed to be on a
straight line are scattered around that line then
one should estimate the correlation between the
two variables. The correlation coefficient serves
as a measure for the correlation of these two
variables. This can be very important if
correlation between results obtained by a new
method and a standard method is required.
r = {nS xiyi – (SxiSyi)}/ {[nSxi2 – (Sxi)2][nSyi2 –
(Syi)2]}1/2
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Example
Calculate the correlation
coefficient of the data :
Solution
First we find Syi2 and
(Syi)2 from the table in
previous example
Syi2 = 2554.66
(Syi)2 = 6988.96
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Substituting in the correlation coefficient equation
above:
r = {5*46.6-(1.500*83.6)} / {[5*0.850 –
2.250][5*2554.66-6988.96]}1/2
r = 1.00
The correlation coefficient occurs between + 1. As
the correlation coefficient approaches unity,
correlation increases and exact correlation
occurs when r = 1. An r value less than 0.90 is
considered bad while that exceeding 0.99 is
considered excellent.
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Currently, many scientists prefer to use the
square of the correlation coefficient, r2
rather than r, to express correlation.
Evidently, the use of r2 is a more strict
criterion as a smaller value is always
obtained when fractions are squared.
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