# 3.9: Problems


1. The following masses were recorded for 12 different U.S. quarters (all given in grams):

 5.683 5.549 5.548 5.552 5.62 5.536 5.539 5.684 5.551 5.552 5.554 5.632

Report the mean, median, range, standard deviation and variance for this data.

2. A determination of acetaminophen in 10 separate tablets of Excedrin Extra Strength Pain Reliever gives the following results (in mg)

 224.3 240.4 246.3 239.4 253.1 261.7 229.4 255.5 235.5 249.7

(a) Report the mean, median, range, standard deviation and variance for this data.

(b) Assuming that $$\overline{X}$$ and s2 are good approximations for $$\mu$$ and for $$\sigma^2$$, and that the population is normally distributed, what percentage of tablets contain more than the standard amount of 250 mg acetaminophen per tablet?

The data in this problem are from Simonian, M. H.; Dinh, S.; Fray, L. A. Spectroscopy 1993, 8(6), 37–47.

3. Salem and Galan developed a new method to determine the amount of morphine hydrochloride in tablets. An analysis of tablets with different nominal dosages gave the following results (in mg/tablet).

100-mg tablets 60-mg tablets 30-mg tablets 10-mg tablets
99.17 54.21 28.51 9.06
94.31 55.62 26.25 8.83
95.92 57.40 25.92 9.08
94.55 57.51 28.62
93.83 52.59 24.93

(a) For each dosage, calculate the mean and the standard deviation for the mg of morphine hydrochloride per tablet.

(b) For each dosage level, and assuming that $$\overline{X}$$ and s2 are good approximations for $$\mu$$ and for $$\sigma^2$$, and that the population is normally distributed, what percentage of tablets contain more than the nominal amount of morphine hydro- chloride per tablet?

The data in this problem are from Salem, I. I.; Galan, A. C. Anal. Chim. Acta 1993, 283, 334–337.

4. Daskalakis and co-workers evaluated several procedures for digesting oyster and mussel tissue prior to analyzing them for silver. To evaluate the procedures they spiked samples with known amounts of silver and analyzed the samples to determine the amount of silver, reporting results as the percentage of added silver found in the analysis. A procedure was judged acceptable if its spike recoveries fell within the range 100±15%. The spike recoveries for one method are shown here.

 105% 108% 92% 99% 101% 93% 93% 104%

Assuming a normal distribution for the spike recoveries, what is the probability that any single spike recovery is within the accepted range?

The data in this problem are from Daskalakis, K. D.; O’Connor, T. P.; Crecelius, E. A. Environ. Sci. Technol. 1997, 31, 2303– 2306. See Chapter 15 to learn more about using a spike recovery to evaluate an analytical method.

5. The formula weight (FW) of a gas can be determined using the following form of the ideal gas law

$FW = \frac {g \text{R} T} {P V} \nonumber$

where g is the mass in grams, R is the gas constant, T is the temperature in Kelvin, P is the pressure in atmospheres, and V is the volume in liters. In a typical analysis the following data are obtained (with estimated uncertainties in parentheses)

g = 0.118 g (± 0.002 g)

R = 0.082056 L atm mol–1 K–1 (± 0.000001 L atm mol–1 K–1)

T = 298.2 K (± 0.1 K)

P = 0.724 atm (± 0.005 atm)

V = 0.250 L (± 0.005 L)

(a) What is the compound’s formula weight and its estimated uncertainty?

(b) To which variable(s) should you direct your attention if you wish to improve the uncertainty in the compound’s molecular weight?

6. To prepare a standard solution of Mn2+, a 0.250 g sample of Mn is dissolved in 10 mL of concentrated HNO3 (measured with a graduated cylinder). The resulting solution is quantitatively transferred to a 100-mL volumetric flask and diluted to volume with distilled water. A 10-mL aliquot of the solution is pipeted into a 500-mL volumetric flask and diluted to volume.

(a) Express the concentration of Mn in mg/L, and estimate its uncertainty using a propagation of uncertainty.

(b) Can you improve the concentration’s uncertainty by using a pipet to measure the HNO3, instead of a graduated cylinder?

7. The mass of a hygroscopic compound is measured using the technique of weighing by difference. In this technique the compound is placed in a sealed container and weighed. A portion of the compound is removed and the container and the remaining material are reweighed. The difference between the two masses gives the sample’s mass. A solution of a hygroscopic compound with a gram formula weight of 121.34 g/mol (±0.01 g/mol) is prepared in the following manner. A sample of the compound and its container has a mass of 23.5811 g. A portion of the compound is transferred to a 100-mL volumetric flask and diluted to volume. The mass of the compound and container after the transfer is 22.1559 g. Calculate the compound’s molarity and estimate its uncertainty by a propagation of uncertainty.

8. Use a propagation of uncertainty to show that the standard error of the mean for n determinations is $$\sigma / \sqrt{n}$$.

9. Beginning with Equation 4.6.4 and Equation 4.6.5, use a propagation of uncertainty to derive Equation 4.6.6.

10. What is the smallest mass you can measure on an analytical balance that has a tolerance of ±0.1 mg, if the relative error must be less than 0.1%?

11. Which of the following is the best way to dispense 100.0 mL if we wish to minimize the uncertainty: (a) use a 50-mL pipet twice; (b) use a 25-mL pipet four times; or (c) use a 10-mL pipet ten times?

12. You can dilute a solution by a factor of 200 using readily available pipets (1-mL to 100-mL) and volumetric flasks (10-mL to 1000-mL) in either one step, two steps, or three steps. Limiting yourself to the glassware in Table 4.2.1, determine the proper combination of glassware to accomplish each dilution, and rank them in order of their most probable uncertainties.

13. Explain why changing all values in a data set by a constant amount will change $$\overline{X}$$ but has no effect on the standard deviation, s.

14. Obtain a sample of a metal, or other material, from your instructor and determine its density by one or both of the following methods:

Method A: Determine the sample’s mass with a balance. Calculate the sample’s volume using appropriate linear dimensions.

Method B: Determine the sample’s mass with a balance. Calculate the sample’s volume by measuring the amount of water it displaces by adding water to a graduated cylinder, reading the volume, adding the sample, and reading the new volume. The difference in volumes is equal to the sample’s volume.

Determine the density at least five times.

(a) Report the mean, the standard deviation, and the 95% confidence interval for your results.

(b) Find the accepted value for the metal’s density and determine the absolute and relative error for your determination of the metal’s density.

(c) Use a propagation of uncertainty to determine the uncertainty for your method of analysis. Is the result of this calculation consistent with your experimental results? If not, suggest some possible reasons for this disagreement.

15. How many carbon atoms must a molecule have if the mean number of 13C atoms per molecule is at least one? What percentage of such molecules will have no atoms of 13C?

16. In Example 4.4.1 we determined the probability that a molecule of cholesterol, C27H44O, had no atoms of 13C.

(a) Calculate the probability that a molecule of cholesterol, has 1 atom of 13C.

(b) What is the probability that a molecule of cholesterol has two or more atoms of 13C?

17. Berglund and Wichardt investigated the quantitative determination of Cr in high-alloy steels using a potentiometric titration of Cr(VI). Before the titration, samples of the steel were dissolved in acid and the chromium oxidized to Cr(VI) using peroxydisulfate. Shown here are the results ( as %w/w Cr) for the analysis of a reference steel.

 16.968 16.922 16.84 16.883 16.887 16.977 16.857 16.728

Calculate the mean, the standard deviation, and the 95% confidence interval about the mean. What does this confidence interval mean?

The data in this problem are from Berglund, B.; Wichardt, C. Anal. Chim. Acta 1990, 236, 399–410.

18. Ketkar and co-workers developed an analytical method to determine trace levels of atmospheric gases. An analysis of a sample that is 40.0 parts per thousand (ppt) 2-chloroethylsulfide gave the following results

 43.3 34.8 31.9 37.8 34.4 31.9 42.1 33.6 35.3

(a) Determine whether there is a significant difference between the experimental mean and the expected value at $$\alpha = 0.05$$.

(b) As part of this study, a reagent blank was analyzed 12 times giving a mean of 0.16 ppt and a standard deviation of 1.20 ppt. What are the IUPAC detection limit, the limit of identification, and limit of quantitation for this method assuming $$\alpha = 0.05$$?

The data in this problem are from Ketkar, S. N.; Dulak, J. G.; Dheandhanou, S.; Fite, W. L. Anal. Chim. Acta 1991, 245, 267–270.

19. To test a spectrophotometer’s accuracy a solution of 60.06 ppm K2Cr2O7 in 5.0 mM H2SO4 is prepared and analyzed. This solution has an expected absorbance of 0.640 at 350.0 nm in a 1.0-cm cell when using 5.0 mM H2SO4 as a reagent blank. Several aliquots of the solution produce the following absorbance values.

 0.639 0.638 0.64 0.639 0.64 0.639 0.638

Determine whether there is a significant difference between the experimental mean and the expected value at $$\alpha = 0.01$$.

20. Monna and co-workers used radioactive isotopes to date sediments from lakes and estuaries. To verify this method they analyzed a 208Po standard known to have an activity of 77.5 decays/min, obtaining the following results.

 77.09 75.37 72.42 76.84 77.84 76.69 78.03 74.96 77.54 76.09 81.12 75.75

Determine whether there is a significant difference between the mean and the expected value at $$\alpha = 0.05$$.

The data in this problem are from Monna, F.; Mathieu, D.; Marques, A. N.; Lancelot, J.; Bernat, M. Anal. Chim. Acta 1996, 330, 107–116.

21. A 2.6540-g sample of an iron ore, which is 53.51% w/w Fe, is dissolved in a small portion of concentrated HCl and diluted to volume in a 250-mL volumetric flask. A spectrophotometric determination of the concentration of Fe in this solution yields results of 5840, 5770, 5650, and 5660 ppm. Determine whether there is a significant difference between the experimental mean and the expected value at $$\alpha = 0.05$$.

22. Horvat and co-workers used atomic absorption spectroscopy to determine the concentration of Hg in coal fly ash. Of particular interest to the authors was developing an appropriate procedure for digesting samples and releasing the Hg for analysis. As part of their study they tested several reagents for digesting samples. Their results using HNO3 and using a 1 + 3 mixture of HNO3 and HCl are shown here. All concentrations are given as ppb Hg sample.

 HNO3: 161 165 160 167 166 1 + 3 HNO3 – HCl: 159 145 154 147 143 156

Determine whether there is a significant difference between these methods at $$\alpha = 0.05$$.

The data in this problem are from Horvat, M.; Lupsina, V.; Pihlar, B. Anal. Chim. Acta 1991, 243, 71–79.

23, Lord Rayleigh, John William Strutt (1842-1919), was one of the most well known scientists of the late nineteenth and early twentieth centuries, publishing over 440 papers and receiving the Nobel Prize in 1904 for the discovery of argon. An important turning point in Rayleigh’s discovery of Ar was his experimental measurements of the density of N2. Rayleigh approached this experiment in two ways: first by taking atmospheric air and removing O2 and H2; and second, by chemically producing N2 by decomposing nitrogen containing compounds (NO, N2O, and NH4NO3) and again removing O2 and H2. The following table shows his results for the density of N2, as published in Proc. Roy. Soc. 1894, LV, 340 (publication 210); all values are the grams of gas at an equivalent volume, pressure, and temperature.

atmospheric origin chemical origin
2.31017 2.30143
2.30986 2.29890
2.31010 2.29816
2.31001 2.30182
2.31024 2.29869
2.31010 2.29940
2.31028 2.29849
2.29889

Explain why this data led Rayleigh to look for and to discover Ar. You can read more about this discovery here: Larsen, R. D. J. Chem. Educ. 1990, 67, 925–928.

24. Gács and Ferraroli reported a method for monitoring the concentration of SO2 in air. They compared their method to the standard method by analyzing urban air samples collected from a single location. Samples were collected by drawing air through a collection solution for 6 min. Shown here is a summary of their results with SO2 concentrations reported in $$\mu \text{L/m}^3$$.

standard method new method
21.62 21.54
22.20 20.51
24.27 22.31
23.54 21.30
24.25 24.62
23.09 25.72
21.02 21.54

Using an appropriate statistical test, determine whether there is any significant difference between the standard method and the new method at $$\alpha = 0.05$$.

The data in this problem are from Gács, I.; Ferraroli, R. Anal. Chim. Acta 1992, 269, 177–185.

25. One way to check the accuracy of a spectrophotometer is to measure absorbances for a series of standard dichromate solutions obtained from the National Institute of Standards and Technology. Absorbances are measured at 257 nm and compared to the accepted values. The results obtained when testing a newly purchased spectrophotometer are shown here. Determine if the tested spectrophotometer is accurate at $$\alpha = 0.05$$.

standard measured absorbance expected absorbance
1 0.2872 0.2871
2 0.5773 0.5760
3 0.8674 0.8677
4 1.1623 1.1608
5 1.4559 1.4565

26. Maskarinec and co-workers investigated the stability of volatile organics in environmental water samples. Of particular interest was establishing the proper conditions to maintain the sample’s integrity between its collection and its analysis. Two preservatives were investigated—ascorbic acid and sodium bisulfate—and maximum holding times were determined for a number of volatile organics and water matrices. The following table shows results for the holding time (in days) of nine organic compounds in surface water.

compound Ascorbic Acid Sodium Bisulfate
methylene chloride 77 62
carbon disulfide 23 54
trichloroethane 52 51
benzene 62 42
1,1,2-trichlorethane 57 53
1,1,2,2-tetrachloroethane 33 85
tetrachloroethene 32 94
chlorbenzene 36 86

Determine whether there is a significant difference in the effectiveness of the two preservatives at $$\alpha = 0.10$$.

The data in this problem are from Maxkarinec, M. P.; Johnson, L. H.; Holladay, S. K.; Moody, R. L.; Bayne, C. K.; Jenkins, R. A. Environ. Sci. Technol. 1990, 24, 1665–1670.

27. Using X-ray diffraction, Karstang and Kvalhein reported a new method to determine the weight percent of kaolinite in complex clay minerals using X-ray diffraction. To test the method, nine samples containing known amounts of kaolinite were prepared and analyzed. The results (as % w/w kaolinite) are shown here.

 actual 5 10 20 40 50 60 80 90 95 found 6.8 11.7 19.8 40.5 53.6 61.7 78.9 91.7 94.7

Evaluate the accuracy of the method at $$\alpha = 0.05$$.

The data in this problem are from Karstang, T. V.; Kvalhein, O. M. Anal. Chem. 1991, 63, 767–772.

28. Mizutani, Yabuki and Asai developed an electrochemical method for analyzing l-malate. As part of their study they analyzed a series of beverages using both their method and a standard spectrophotometric procedure based on a clinical kit purchased from Boerhinger Scientific. The following table summarizes their results. All values are in ppm.

Sample Electrode Spectrophotometric
Apple Juice 1 34.0 33.4
Apple Juice 2 22.6 28.4
Apple Juice 3 29.7 29.5
Apple Juice 4 24.9 24.8
Grape Juice 1 17.8 18.3
Grape Juice 2 14.8 15.4
Mixed Fruit Juice 1 8.6 8.5
Mixed Fruit Juice 2 31.4 31.9
White Wine 1 10.8 11.5
White Wine 2 17.3 17.6
White Wine 3 15.7 15.4
White Wine 4 18.4 18.3

The data in this problem are from Mizutani, F.; Yabuki, S.; Asai, M. Anal. Chim. Acta 1991, 245,145–150.

29. Alexiev and colleagues describe an improved photometric method for determining Fe3+ based on its ability to catalyze the oxidation of sulphanilic acid by KIO4. As part of their study, the concentration of Fe3+ in human serum samples was determined by the improved method and the standard method. The results, with concentrations in $$\mu \text{mol/L}$$, are shown in the following table.

Sample Improved Method Standard Method
1 8.25 8.06
2 9.75 8.84
3 9.75 8.36
4 9.75 8.73
5 10.75 13.13
6 11.25 13.65
7 13.88 13.85
8 14.25 13.43

Determine whether there is a significant difference between the two methods at $$\alpha = 0.05$$.

The data in this problem are from Alexiev, A.; Rubino, S.; Deyanova, M.; Stoyanova, A.; Sicilia, D.; Perez Bendito, D. Anal. Chim. Acta, 1994, 295, 211–219.

30. Ten laboratories were asked to determine an analyte’s concentration of in three standard test samples. Following are the results, in $$\mu \text{g/ml}$$.

Laboratory Sample 1 Sample 2 Sample 3
1 22.6 13.6 16.0
2 23.0 14.9 15.9
3 21.5 13.9 16.9
4 21.9 13.9 16.9
5 21.3 13.5 16.7
6 22.1 13.5 17.4
7 23.1 13.5 17.5
8 21.7 13.5 16.8
9 22.2 13.0 17.2
10 21.7 13.8 16.7

Determine if there are any potential outliers in Sample 1, Sample 2 or Sample 3. Use all three methods—Dixon’s Q-test, Grubb’s test, and Chauvenet’s criterion—and compare the results to each other. For Dixon’s Q-test and for the Grubb’s test, use a significance level of $$\alpha = 0.05$$.

The data in this problem are adapted from Steiner, E. H. “Planning and Analysis of Results of Collaborative Tests,” in Statistical Manual of the Association of Official Analytical Chemists, Association of Official Analytical Chemists: Washington, D. C., 1975.

31.When copper metal and powdered sulfur are placed in a crucible and ignited, the product is a sulfide with an empirical formula of CuxS. The value of x is determined by weighing the Cu and the S before ignition and finding the mass of CuxS when the reaction is complete (any excess sulfur leaves as SO2). The following table shows the Cu/S ratios from 62 such experiments (note that the values are organized from smallest-to-largest by rows).

 1.764 1.838 1.865 1.866 1.872 1.877 1.89 1.891 1.891 1.897 1.899 1.900 1.906 1.908 1.910 1.911 1.916 1.919 1.92 1.922 1.927 1.931 1.935 1.936 1.936 1.937 1.939 1.939 1.940 1.941 1.941 1.942 1.943 1.948 1.953 1.955 1.957 1.957 1.957 1.959 1.962 1.963 1.963 1.963 1.966 1.968 1.969 1.973 1.975 1.976 1.977 1.981 1.981 1.988 1.993 1.993 1.995 1.995 1.995 2.017 2.029 2.042

(a) Calculate the mean, the median, and the standard deviation for this data.

(b) Construct a histogram for this data. From a visual inspection of your histogram, do the data appear normally distributed?

(c) In a normally distributed population 68.26% of all members lie within the range $$\mu \pm 1 \sigma$$. What percentage of the data lies within the range $$\overline{X} \pm 1 \sigma$$? Does this support your answer to the previous question?

(d) Assuming that $$\overline{X}$$ and $$s^2$$ are good approximations for $$\mu$$ and for $$\sigma^2$$, what percentage of all experimentally determined Cu/S ratios should be greater than 2? How does this compare with the experimental data? Does this support your conclusion about whether the data is normally distributed?

(e) It has been reported that this method of preparing copper sulfide results in a non-stoichiometric compound with a Cu/S ratio of less than 2. Determine if the mean value for this data is significantly less than 2 at a significance level of $$\alpha = 0.01$$.

See Blanchnik, R.; Müller, A. “The Formation of Cu2S From the Elements I. Copper Used in Form of Powders,” Thermochim. Acta, 2000, 361, 31-52 for a discussion of some of the factors affecting the formation of non-stoichiometric copper sulfide. The data in this problem were collected by students at DePauw University.

32. Real-time quantitative PCR is an analytical method for determining trace amounts of DNA. During the analysis, each cycle doubles the amount of DNA. A probe species that fluoresces in the presence of DNA is added to the reaction mixture and the increase in fluorescence is monitored during the cycling. The cycle threshold, $$C_t$$, is the cycle when the fluorescence exceeds a threshold value. The data in the following table shows $$C_t$$ values for three samples using real-time quantitative PCR. Each sample was analyzed 18 times.

Sample X Sample Y Sample Z
24.24 25.14 24.41 28.06 22.97 23.43
23.97 24.57 27.21 27.77 22.93 23.66
24.44 24.49 27.02 28.74 22.95 28.79
24.79 24.68 26.81 28.35 23.12 23.77
23.92 24.45 26.64 28.80 23.59 23.98
24.53 24,48 27.63 27.99 23.37 23.56
24.95 24.30 28.42 28.21 24.17 22.80
24.76 24.60 25.16 28.00 23.48 23.29
25.18 24.57 28.53 28.21 23.80 23.86

Examine this data and write a brief report on your conclusions. Issues you may wish to address include the presence of outliers in the samples, a summary of the descriptive statistics for each sample, and any evidence for a difference between the samples.

The data in this problem is from Burns, M. J.; Nixon, G. J.; Foy, C. A.; Harris, N. BMC Biotechnol. 2005, 5:31 (open access publication).

3.9: Problems is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Harvey.