## Transcribed Text

inter
90% confidence
is = 12.5.
VII
The confidence interval
many
time
90 % repeat sethelude H. u Nine-ter
1
expected are If we expected to include to the true mean, measured
on
inter
1
20
14
TEST YOURSELF six results, wt%) confidence
,0888°
Hydro
Using Vo = 12.50 + 0.31
.007 91
5
4
TABLE 4-4
Values of Student's t
Confidence level (%)
99
98
95
Degrees of freedom
50
90
6.314
12.706
31.821
63.656
1.000
2.920
4.303
6.965
9.925
I
2
0.816
3.182
4.541
5.841
0.765
2.353
3
0.741
2.132
2.776
3.747
4.604
4
2.015
2.571
3.365
4.032
5
0.727
6
0.718
1.943
2.447
3.143
3.707
7
0.711
1.895
2.365
2.998
3.500
8
0.706
1.860
2.306
2.896
3.355
9
0.703
1.833
2.262
2.821
3.250
10
0.700
1.812
2.228
2.764
15
3.169
0.691
1.753
2.131
20
2.602
0.687
1.725
2.947
25
2.086
0.684
2.528
1.708
30
2.060
2.845
0.683
40
1.697
2.485
2.042
2.787
0.681
1.684
2.457
60
0.679
2.021
2.750
120
1.671
2.423
0.677
2.000
2.704
80
1.658
1.980
2.390
0.674
1.645
calculating
confidence
1.960
2.358
2.660
In
population
intervals,
standard
o
may
be
2.326
2.617
deviation.
If
substituted
o
is
used
for
S
instead
in
Values
of
1
in
this
table
of
5,
the
Equation
apply
value
4-7
to
two-tailed
of
if
1
you
to
have
2.576
tailed
use
in
a
great
deal
test, we use values of I listed for 90% tests confidence. illustrated Each in Figure wing outside 4.9a. The of t Equation for 95% confidence 4-7 comes level from of the experience bottom row with of a
The 90% confidence contains specifies 5% this particular table, 2.5% method of the a
standard dom from a Gaussian meaning of Interval confider
Figure Meaning 4-5 illustrates of a Confidence
curve,
their mean
deviation
(a)
population
X =
Ux = k + n 1 + m2 -
(y - V)2 2
=
(4-27)
(C
(xi X) 2
15
V
av
where Sy is the standard deviation of y (Equation 4-20), Iml is the absolute value of the slope (=
X
=
av
ABS(m) in Excel), k is the number of replicate measurements of the unknown, n is the number of
data points for the calibration line (14 in Table 4-8), y is the mean value of y for the points on the
calibration line, Xi are the individual values of X for the points on the calibration line, and X is the
mean value of X for the points on the calibration line. For a single measurement of the unknown,
k = 1 and Equation 4-27 gives Ux = +0.39 g. Four replicate unknowns (k = 4) with an average
corrected absorbance of 0.302 reduce the uncertainty to Ux = +0.23 g.
The confidence interval for X is where t is Student's t (Table 4-4) for n - 2 degrees
To
find
y
of freedom. If Ux = 0.23 g and n = 14 points (12 degrees of freedom), the 95% confidence
use
the
interval for X is ttux = +(2.179)(0.23) = +0.50 g. There is no 1/Vn in the expression for
confidence interval because Ux is the standard deviation of the mean.
TINV(0.0
Propagation of Uncertainty
You now have all the tools required for a more rigorous discussion of propagation of uncertainty
than we had in Chapter 3. If you are so inclined, you will find that discussion in Appendix B.
4-9
A Spreadsheet for Least Squares
Figure 4-15 implements least-squares analysis, including propagation of error with Equation
4-27. Enter values of X and y in columns B and C. Then select cells B10:C12. Enter the for-
mula "=LINEST(C4:C7,B4:B7,TRUE,TRUE)" and press CONTROL + SHIFT + ENTER on
a PC or COMMAND(% + RETURN on a Mac. LINEST returns m, b, Um, Ub, R², and Sy in
A
B
C
D
E
1
F
Least-Squares Spreadsheet
G
2
3
Highlight cells B10:C12
x
y
6
4
Type "= LINEST(C4:C7,
1
2
5
B4:B7,TRUE,TRUE)
y = 0.6154x + 1.3462
3
3
6
5
For PC, press
4
4
7
CTRL+SHIFT+ENTER
6
5
8
For Mac, press
4
9
COMMAND+RETURN
LINEST output:
10
m
0.6154
1.3462
11
b
3
Um
0.0544
12
0.2141
ub
R²
0.9846
13
0.1961
Sy
2
14
n =
4
15
B14 = COUNT(B4:B7)
Mean y =
16
3.5
(x) - mean x) 2 =
B15 = AVERAGE(C4:C7
1
17
13
B16 = DEVSQ(B4:B7)
18
Measured y =
2.72
Input
o
1
19
k = Number of replicate
2
3
4
measurements of y =
x
20
1
Input
21
Derived x =
2.2325
B20 = (B18-C10)/B10
Ux =
FIGURE 4-15 Spreadsheet for linear least-squares 0.3735 B21 analysis. = (C12/ABS(B10)) SQRT ((1/B19)+(1/B14). +
4-9
Spreader
fercury be many times greater than that of water. Contamination of fish by mercury is a problem
is absorbed by fish when they filter water through their gills, and the concentration of mercury because in fish it
an neat have can negative effects on the brain, especially in developing children. High levels of mercury can also have
legative health impacts on adults.
A scientist working for the environmental protection agency (EPA) is working to analyze mercury levels in L fish of
believed to be contaminated. She digests 51.35 grams of tuna fish and dilutes it to a total volume of 1.000
She then uses atomic absorbance spectrophotometry to determine the concentration of the sample. She
uses water. an external calibration method where she analyzes four standards containing mercury at increasing
concentrations, a blank, and the unknown fish sample (one analysis on each sample) and gets the following
results:
Concentration
of mercury
Sample
(ppm)
absorbance
Std 1
5.00
0.602
Std 2
3.00
0.380
Std 3
1.00
0.167
Std 4
0.50
0.127
blank
.020
Unknown
0.297
Your job is to analyze her data using Microsoft Excel, and use it to get the information required on the next
sheet.
1 -
Hint: A note about significant figures. As a general rule, the uncertainty (in this case the standard
deviation) should be reported with 1 or 2 significant figures. The measurement number associated with
that uncertainty can then be rounded to the same decimal place (not the same # of significant figures) as
the uncertainty. For example, if m (the slope of the calib curve) = 2.3456 and Sm (the uncertainty in the
slope)
= 0.04253, then the proper way to express the slope with uncertainty rounded to 2 significant
figures would be: 2.346 0.042. It would also be acceptable to round the uncertainty to only 1
significant figure, i.e. 2.35 I 0.04. I will accept answers where the uncertainty is rounded to 1 or 2
significant figures, just be consistent and be sure that the measurement number matches the decimal of
the uncertainty. It is NOT acceptable to use more than 2 significant figures in the uncertainty.
1. Slope of calibration curve (m)
=
2. Uncertainty of slope (Sm)
=
3. Intercept of calibration curve (b)
=
4. Uncertainty of intercept (Sb)
=
5. Concentration of mercury in the water sample (ppm)
=
6. Uncertainty in concentration of mercury in water (Sx) (ppm)=
7. Concentration of mercury in the tuna (in mg/kg)
=
8. Uncertainty in conc. of mercury in the tuna (mg/kg)
=
9. 95% confidence interval for concentration of mercury in tuna (mg/kg)* =
ote: To calculate the 95% confidence interval from calibration data, use the equation = tSx, where Sx is the
certainty you calculated for the concentration of tuna in mg/kg, and t is for N-2 degrees
of
freedom
(where
N
the number of data points in the calibration curve). In other situations we have calculated a confidence
erval with the equation ts/Nn. However, the equation that calculates the uncertainty in X (equation 4-27 of
rris 9th edition) already divides by Vn, therefore, it is not necessary to include this Vn term in your calculation
the confidence interval when you use calibration data. Note that N-2 degrees of freedom (rather than N-1
grees of freedom) is used because you have both X and y data.

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.