Chapter 2: Supplementary Exercises
2.S.1. A Public Analyst’s laboratory routinely measures potable water samples by flame atomic absorption spectrometry to ensure compliance with the EU drinking water directive [1]. One of the parameters that is measured is the zinc concentration. Every time a batch of test samples is measured the batch includes two replicates of a QC control sample spiked with 200 g L-1 of zinc. The replicates are averaged to give a reported concentration. The data are presented in Table 2.23 below.
Run Rep-1 Rep-2 Conc. Run Rep-1 Rep-2 Conc.
1 194.00 197.00 195.50 16 201.00 200.00 200.50 2 195.00 195.00 195.00 17 191.00 194.00 192.50 3 202.00 203.00 202.50 18 201.00 207.00 204.00 4 204.00 209.00 206.50 19 193.00 192.00 192.50 5 202.00 200.00 201.00 20 196.00 200.00 198.00 6 195.00 195.00 195.00 21 198.00 195.00 196.50 7 192.00 195.00 193.50 22 192.00 199.00 195.50 8 194.00 193.00 193.50 23 191.00 192.00 191.50 9 202.00 200.00 201.00 24 192.00 195.00 193.50 10 192.00 191.00 191.50 25 185.00 185.00 185.00 11 200.00 204.00 202.00 26 188.00 192.00 190.00 12 190.00 188.00 189.00 27 188.00 194.00 191.00 13 184.00 184.00 184.00 28 191.00 188.00 189.50 14 197.00 196.00 196.50 29 200.00 200.00 200.00 15 198.00 197.00 197.50 30 203.00 206.00 204.50
Table 2.S.1.1: Replicate zinc measurements on spiked water samples
Summary statistics for the concentrations are given below.
Variable N Mean StDev
Conc. 30 195.62 5.62
Table 2.S.1.2: Summary statistics for the zinc data
The assumptions underlying t-tests and their corresponding confidence intervals require that the data are independent of each other and come from a single stable Normal distribution. Two pictures are presented overleaf which may be used to assess these assumptions – a Normal plot and a control chart for the reported concentration results. The laboratory uses two rules for signalling out-of-control conditions: any point outside the control limits and a run of nine points above or below the centre line. Are you satisfied that the assumptions are valid in this case? Carry out a t-test of the hypothesis that the concentrations vary around a long-run average value of 200 g L-1. Calculate and interpret a 95% confidence interval for the mean around which the
measurement system is varying. Explain the relationship between the two statistical methods you have used: the t-test and the confidence interval.
210 205 200 195 190 185 180 2 1 0 -1 -2 C o n c. Score Mean 195.6 StDev 5.623 N 30 AD 0.203 P-Value 0.864
Figure 2.S.1.1: Normal plot of zinc concentration data
28 25 22 19 16 13 10 7 4 1 210 205 200 195 190 185 180 Observation In d iv id u a l V a lu e _ X=195.62 UCL=210.57 LCL=180.67
2.S.2. Samuels [2] cites a study by Golden et. al. [3] on brain damage due to alcoholism. She presents the data shown in Table 2.S.2.1 which represent brain densities (measurement units not given) measured by CT scanning for 11 chronic alcoholics and 11 controls. The controls were individually matched to the alcoholics on age, sex, education and other variables. Table 2.S.2.2 gives some summary statistics for the study.
Alcoholics Controls differences
40.1 41.3 1.2 38.5 40.2 1.7 36.9 37.4 0.5 41.4 46.1 4.7 40.6 43.9 3.3 42.3 41.9 -0.4 37.2 39.9 2.7 38.6 40.4 1.8 38.5 38.6 0.1 38.4 38.1 -0.3 38.1 39.5 1.4
Table 2.S.2.1: Brain densities measured by CT scans
Variable N Mean SE Mean StDev
Alcoholics 11 39.145 0.519 1.722 Controls 11 40.664 0.773 2.563 differences 11 1.518 0.476 1.580 Table 2.S.2.2: Descriptive statistics for brain density study
It is of interest to compare the densities for the two groups of subjects. Are their averages statistically significantly different? If yes, how different? Does the assumption (underlying a paired t-test) of data Normality appear reasonable in this case?
2.S.3. Table 2.S.3.1 shows data taken from Moore [4] referring to the absorption of drugs in the blood of 20 healthy non-smoking male subjects for a pair of reference and generic drugs. Units are not given in the textbook. Numbers 1-20 were assigned at random to the subjects and then subjects 1-10 received the generic drug first, 11-20 the reference drug first. There was a washout period between the administrations of the two drugs so that there would not be a carryover effect.
Subject Generic Reference Difference
1 3340 1735 1605 2 2613 2594 19 3 1138 2526 -1388 4 2738 2344 394 5 1287 938 349 6 1284 1022 262 7 1930 1339 591 8 2120 2463 -343 9 1613 2779 -1166 10 3052 2256 796 11 2549 1438 1111 12 1310 1833 -523 13 2254 3852 -1598 14 1964 1262 702 15 1755 4108 -2353 16 2302 1864 438 17 1682 1829 -147 18 1851 2059 -208 19 3050 1020 2030 20 1878 1709 169
Table 2.S.3.1: Generic-reference drug data
Variable N Mean StDev
Generic 20 2086 643 Reference 20 2049 860 Difference 20 37.0 1071
Table 2.S.3.2: Summaries for generic-reference drug data
It is of interest to know whether the absorption rate is different for the generic drug. Analyse the data and draw conclusions.
2.S.4. Moore [4] presents the following IQ scores for seventh-grade girls and boys in a US Mid-West school district. If this can be regarded as a random sample from all seventh-grade children in the district, is there evidence of a difference in IQ scores for girls and boys? Ignore the Boys-all column (to avoid having to deal with unequal sample sizes in carrying out the test).
Girls Boys Boys-all
114 111 111 112 108 109 109 127 111 124 124 110 100 93 93 105 130 90 90 104 103 107 107 136 104 113 113 120 120 126 126 105 74 123 123 106 89 114 114 106 132 100 100 103 112 128 128 110 102 116 116 113 111 79 79 123 107 106 106 115 91 107 107 77 128 128 128 103 127 127 114 119 119 118 115 115 98 118 118 114 119 119 119 110 110 96 111 111 103 113 113 86 97 97 112 110 110 105 97 97 72 124 124 112 102 102 93 107 107
Table 2.S.4.1: Moore’s IQ data
Variable N Mean StDev
Girls 31 105.84 14.27 Boys 31 111.06 11.89 Boys-all 47 110.96 12.12
2.S.5 Epidermolysis Bullosa (EB) is a rare genetic skin disease affecting approximately 1 in 17,000 people. Those suffering from the form of the disease known as dominant dystrophic EB (dDEB) have mutations in the gene known as COL7A1. The function of this gene is to produce collagen type VII protein, which anchors the different layers of skin together. Suffers produce a mutant form of the protein, which causes their skin to be extremely fragile and to blister when the skin surface is rubbed.
A molecule (here labeled T for target) designed to suppress the COL7A1 gene was delivered to human keratinocyte skin cells. As a control a second molecule, C, was also delivered to cells (this molecule did not target the COL7A1 gene). There were 18 replicates for each molecule type. The two sets of cells were subsequently analysed to measure the amounts of COL7A1 gene product produced.
Table 2.S.5.1. shows the resulting data1: the responses, derived from an instrumental assay, are dimensionless – they may be transformed to natural units via a calibration line.
T C 0.572 0.995 0.820 1.082 0.560 0.883 0.352 1.007 0.458 1.125 0.413 1.074 0.464 0.843 0.693 1.089 0.625 1.217 0.221 0.929 0.257 1.128 0.218 0.855 0.330 0.904 0.511 1.137 0.474 0.943 0.621 1.083 0.430 1.097 0.430 1.167 Mean 0.469 1.031 St. Dev. 0.163 0.114
Table 2.S.5.1: Amounts of collagen type VII protein produced in the study
The question of interest here is whether the T molecule results in a lowering of the amount of collagen type VII protein produced, relative to the control molecule, C.
