Question

Question 4. (20 pts.) The objectives of a study by Skjelbo et al. were to examine (a) the relationship between Chloroguanide metabolism and efficacy in malaria prophylaxis and (b) the mephenytoin metabolism and its relationship to chloroguanide metabolism among Tanzanians. From information provided by urine specimens from the 216 subjects, the investigators computed the ratio of unchanged S-mephenytoin to R-mephenytoin (S/R ratio). The results were as follows: 0.0269 0.0400 0.0550 0.0550 0.0650 0.0670 0.0700 0.0720 0.0760 0.0850 0.0870 0.0870 0.0880 0.0900 0.0900 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.0990 0.1000 0.1020 0.1040 0.1050 0.1050 0.1080 0.1080 0.1090 0.1090 0.1090 0.1160 0.1190 0.1200 0.1230 0.1240 0.1340 0.1340 0.1370 0.1390 0.1460 0.1480 0.1490 0.1490 0.1500 0.1500 0.1500 0.1540 0.1550 0.1570 0.1600 0.1650 0.1650 0.1670 0.1670 0.1677 0.1690 0.1710 0.1720 0.1740 0.1780 0.1780 0.1790 0.1790 0.1810 0.1880 0.1890 0.1890 0.1920 0.1950 0.1970 0.2010 0.2070 0.2100 0.2100 0.2140 0.2150 0.2160 0.2260 0.2290 0.2390 0.2400 0.2420 0.2430 0.2450 0.2450 0.2460 0.2460 0.2470 0.2540 0.2570 0.2600 0.2620 0.2650 0.2650 0.2680 0.2710 0.2800 0.2800 0.2870 0.2880 0.2940 0.2970 0.2980 0.2990 0.3000 0.3070 0.3100 0.3110 0.3140 0.3190 0.3210 0.3400 0.3440 0.3480 0.3490 0.3520 0.3530 0.3570 0.3630 0.3630 0.3660 0.3830 0.3900 0.3960 0.3990 0.4080 0.4080 0.4090 0.4090 0.4100 0.4160 0.4210 0.4260 0.4290 0.4290 0.4300 0.4360 0.4370 0.4390 0.4410 0.4410 0.4430 0.4540 0.4680 0.4810 0.4870 0.4910 0.4980 0.5030 0.5060 0.5220 0.5340 0.5340 0.5460 0.5480 0.5480 0.5490 0.5550 0.5920 0.5930 0.6010 0.6240 0.6280 0.6380 0.6600 0.6720 0.6820 Mth251 Page 2 (a) From these data construct the following distributions: frequency, relative frequency, cumulative frequency, and cumulative relative frequency; and the following graphs: histogram, frequency polygon. (b) The investigators defined as poor metabolizers of mephenytoin any subject with an S/R mephenytoin ratio greater than .9. How many and what percentage of the subjects were poor metabolizers? (c) How many and what percentage of the subjects had ratios less than 7? Between .3 and .6999 inclusive?

The objectives of a study by Skjelbo et al. were to examine (a) the relationship between Chloroguanide metabolism and efficacy in malaria prophylaxis and (b) the mephenytoin metabolism and its relationship to chloroguanide metabolism among Tanzanians. From information provided by urine specimens from the 216 subjects, the investigators computed the ratio of unchanged S-mephenytoin to R-mephenytoin (S/R ratio). The results were as follows:

Answer 1

Solution:

a) For calculating frequency, relative frequency, cumulative frequency and cumulative relative frequency we use Ms-Excel and formulae for these are as follows,

• Frequency: The number of times a value of the data occurs.
• Relative Frequency: The ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes.
• Cumulative Frequency: The Cumulative Frequency is the sum of the frequencies for all values that are less than or equal to the given value.
• Cumulative Relative Frequency: The term applies to an ordered set of observations from smallest to largest. The Cumulative Relative Frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.

Also for drawing frequency polygon and histogram we use Ms-Excel.

I have attached the images of all above calculations.

b) By observing all the data, none of the S/R ratio is greater than 0.9.

Book1 - Excel Sign in 0 X File Home Insert Page Layout Formulas Data Review View Help Tell me what you want to do & Share Σ AutoSum Calibri AA ab Wrap Text General Normal Bad & Cut Es Copy Format Painter <K> ATO Fill - Paste BIU- Α και Merge & Center 0 % .00 20 Neutral Insert Delete Format .00 Conditional Format as Good Formatting Table Styles Clear Sort & Find & Filter - Select - Editing Clipboard Font Alignment Number Cells Y1 *** X fx G H — J K L M N O P Q R S T U V w X Y х - freq 1 Histogram 1 1 1 45 2 40 1 ܢܙ 35 1 30 1 ܢܙ 25 0.0269 0.04 0.055 0.065 0.067 0.07 0.072 0.076 0.085 0.087 0.088 0.09 0.099 1 20 1 1 1 15 2 10 1 1 5 2 0 30 A B с E 1 Row Labels - Count of x Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 20.0269 1 0.005208333 1 0.005208333 30.04 1 0.005208333 2 0.010416667 4 0.055 2 2 0.010416667 4 0.020833333 5 0.065 1 0.005208333 5 0.026041667 6 0.067 1 1 0.005208333 6 0.03125 7 0.07 1 0.005208333 7 0.036458333 8 0.072 1 1 0.005208333 8 0.041666667 9 0.076 1 1 0.005208333 9 0.046875 10 0.085 1 0.005208333 10 0.052083333 11 0.087 2 2 0.010416667 12 0.0625 12 0.088 1 0.005208333 13 0.067708333 13 0.09 2 2 0.010416667 15 0.078125 14 0.099 30 30 0.15625 45 0.234375 15 0.1 1 1 0.005208333 46 0.239583333 16 0.102 1 1 0.005208333 47 0.244791667 17 0.104 1 1 0.005208333 48 0.25 18 0.105 2 2 0.010416667 50 0.260416667 19 0.108 2 2 0.010416667 52 0.270833333 20 0.109 3 3 0.015625 55 0.286458333 21 0.116 1 1 0.005208333 56 0.291666667 22 0.119 1 0.005208333 57 0.296875 23 0.12 1 1 0.005208333 58 0.302083333 24 0.123 1 0.005208333 59 0.307291667 25 0.124 1 1 0.005208333 60 0.3125 26 0.134 2 2 0.010416667 62 0.322916667 27 0.137 1 1 0.005208333 63 0.328125 28 0.139 1 0.005208333 64 0.333333333 29 0.146 1 1 0.005208333 65 0.338541667 (0.1369, 0.2469] (0.3569, 0.4669] (0.5769, 0.6869) [0.0269, 0.1369] (0.2469, 0.3569] (0.4669, 0.5769] 1 1 1 Frequency Polygon 2 2 35 3 30 1 25 1 0.1 0.102 0.104 0.105 0.108 0.109 0.116 0.119 0.12 0.123 0.124 0.134 0.137 0.139 0.146 1 20 1 15 1 1 1 10 5 2 1 0 1 1 0.0269 200 0.088 0.104 0.119 0.137 0.15 0.165 0.172 0.188$ 0.201 0.216 0.242 0.254 0.268 0.294 0.307 0.321 0.352 0.383 0.409 0.429 0.441 0.487 0.522 0.555 0.628 1

Book1 - Excel Sign in X File Home Insert Page Layout Formulas Data Review View Help Tell me what you want to do & Share X Cut Calibri Σ AutoSum 11 Å Ä ab Wrap Text lili General Normal ED Bad Eg Copy ATO ΚΙΣΙΣ Fill- Paste BIU A - == % Merge & Center .00 0 Insert Delete Format .00 Neutral Format Painter Conditional Format as Good Formatting Table Styles Clear Sort & Find & Filter - Select Editing Clipboard Font Alignment Number Cells Y1 A B C E F G Н. J L K N M O P — Q R S T U V w X Y D 0.005208333 30 0.148 1 1 66 0.34375 1 0.148 0.149 freq 2 2 68 2 3 3 71 0.15 3 1 1 72 1 1 1 73 1 1 1 74 1 31 0.149 32 0.15 33 0.154 34 0.155 35 0.157 36 0.16 37 0.165 38 0.167 39 0.1677 40 0.169 1 1 75 1 2 2 77 2 2 2 79 2 1 1 80 1 1 1 81 1 1 1 82 1 41 0.171 42 0.172 1 1 83 1 1 1 84 1 2 2 86 2 0.010416667 0.015625 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.010416667 0.005208333 0.005208333 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.354166667 0.369791667 0.375 0.380208333 0.385416667 0.390625 0.401041667 0.411458333 0.416666667 0.421875 0.427083333 0.432291667 0.4375 0.447916667 0.458333333 0.463541667 0.46875 0.479166667 0.484375 0.489583333 0.494791667 0.5 0.505208333 0.515625 0.520833333 0.526041667 0.53125 0.536458333 2 2 88 2 0.154 0.155 0.157 0.16 0.165 0.167 0.1677 0.169 0.171 0.172 0.174 0.178 0.179 0.181 0.188 0.189 0.192 0.195 0.197 0.201 0.207 0.21 0.214 0.215 0.216 0.226 1 1 89 1 1 1 90 1 2 92 2 2 1 1 93 1 1 43 0.174 44 0.178 45 0.179 46 0.181 47 0.188 48 0.189 49 0.192 50 0.195 51 0.197 52 0.201 53 0.207 54 0.21 55 0.214 56 0.215 57 0.216 58 0.226 1 94 1 1 1 95 1 1 1 96 1 1 1 97 1 2 2 2 99 100 1 1 1 1 101 1 1 1 1 1 1 102 103 1 1

Book1 - Excel Sign in File Home Insert Page Layout Formulas Data Review View Help Tell me what you want to do Share Σ AutoSum Calibri AA ab Wrap Text General Normal Bad & Cut Es Copy Format Painter ΚΙΣΙΙΣ AY O Fill - Paste BIU ♡ A- Merge & Center 0 % .00 20 Neutral Insert Delete Format .00 Conditional Format as Good Formatting Table Styles Clear Sort & Find & Filter - Select Editing Clipboard Font Alignment Number Cells Y1 ** fx A B C E F G H J K — L M N O Р Q R S T U V W X Y 1 0.226 1 1 1 1 103 104 1 1 1 105 1 1 1 106 1 1 1 107 1 1 1 1 108 110 2 2 2 2 2 2 112 1 1 1 1 113 114 1 1 1 115 1 1 1 1 116 1 0.536458333 0.541666667 0.546875 0.552083333 0.557291667 0.5625 0.572916667 0.583333333 0.588541667 0.59375 0.598958333 0.604166667 0.609375 0.619791667 0.625 0.630208333 0.640625 0.645833333 0.651041667 0.65625 0.661458333 0.666666667 0.671875 1 1 117 1 2 119 58 0.226 59 0.229 60 0.239 61 0.24 62 0.242 63 0.243 64 0.245 65 0.246 66 0.247 67 0.254 68 0.257 69 0.26 70 0.262 71 0.265 72 0.268 73 0.271 74 0.28 75 0.287 76 0.288 77 0.294 78 0.297 79 0.298 80 0.299 81 0.3 82 0.307 83 0.31 84 0.311 85 0.314 86 0.319 2 D 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.005208333 0.005208333 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 2 1 1 120 1 0.229 0.239 0.24 0.242 0.243 0.245 0.246 0.247 0.254 0.257 0.26 0.262 0.265 0.268 0.271 0.28 0.287 0.288 0.294 0.297 0.298 0.299 0.3 0.307 0.31 0.311 0.314 0.319 1 1 121 1 123 2 2 1 2 1 124 1 1 125 1 1 1 1 126 1 1 1 127 1 1 1 128 1 1 1 1 129 130 1 1 0.677083333 1 1 1 1 1 131 132 1 1 1 1 133 0.682291667 0.6875 0.692708333 0.697916667 0.703125 1 1 1 1 134 135 1 1 1

Book1 - Excel Sign in File Home Insert Page Layout Formulas Data Review View Help Tell me what you want to do & Share X Cut Calibri Σ AutoSum 11 Å Ä ab Wrap Text General Bad ED ÂY O Eg Copy ΚΙΣΙΙΣ Fill- Paste BIU- A - % Merge & Center - .00 0 .00 Normal Conditional Format as Good Formatting Table Styles Insert Delete Format Neutral Format Painter Clear Sort & Find & Filter - Select Editing Clipboard Font Alignment Number Cells Y1 X B C E F G H L K M z O — Р Q R S T U w X Y > 1 1 136 1 1 1 137 1 1 1 1 138 1 1 139 0.321 0.34 0.344 0.348 0.349 0.352 0.353 0.357 1 1 1 1 140 1 1 141 1 1 1 1 142 1 1 143 1 2 2 2 145 146 0.363 0.366 1 1 1 1 1 147 0.383 1 1 1 148 1 1 1 149 1 1 1 150 1 A 87 0.321 88 0.34 89 0.344 90 0.348 91 0.349 92 0.352 93 0.353 94 0.357 95 0.363 96 0.366 97 0.383 98 0.39 99 0.396 100 0.399 101 0.408 102 0.409 103 0.41 104 0.416 105 0.421 106 0.426 107 0.429 108 0.43 109 0.436 110 0.437 111 0.439 112 0.441 113 0.443 114 0.454 115 0.468 D 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.005208333 0.005208333 0.005208333 2 2 2 2 0.708333333 0.713541667 0.71875 0.723958333 0.729166667 0.734375 0.739583333 0.744791667 0.755208333 0.760416667 0.765625 0.770833333 0.776041667 0.78125 0.791666667 0.802083333 0.807291667 0.8125 0.817708333 0.822916667 0.833333333 0.838541667 0.84375 0.848958333 0.854166667 0.864583333 0.869791667 0.875 0.880208333 152 154 2 0.39 0.396 0.399 0.408 0.409 0.41 0.416 0.421 0.426 2 1 1 155 1 1 1 156 1 1 1 157 1 1 1 158 1 2 2 160 2 0.429 0.43 1 1 161 1 1 1 1 162 163 1 1 1 1 1 1 164 166 2 2 0.436 0.437 0.439 0.441 0.443 0.454 0.468 1 1 167 2 1 1 1 1 1 168 1 1 169

Book1 - Excel Sign in > File Home Insert Page Layout Formulas Data Review View Help Tell me what you want to do Share * Cut Σ AutoSum Calibri 11 A À III 9 ab Wrap Text General Bad ATO HH Fill - Paste e Copy Format Painter BIU- MA Merge & Center C%) €. .00 .00 0 Normal Conditional Format as Good Formatting Table Styles Neutral KIK Insert Delete Format Clear - Sort & Find & Filter - Select Editing Clipboard Font Alignment IS Number Cells Y1 X fr A B с D E F G - J K M z O Р Q R S T U V w I X Y 1 1 169 1 1 1 1 1 1 170 171 172 1 1 1 0.468 0.481 0.487 0.491 0.498 0.503 0.506 1 1 1 1 1 1 1 1 1 1 173 174 175 176 178 179 1 1 0.522 1 2 2 1 1 2 1 2 1 2 181 2 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.010416667 0.005208333 0.010416667 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.005208333 0.880208333 0.885416667 0.890625 0.895833333 0.901041667 0.90625 0.911458333 0.916666667 0.927083333 0.932291667 0.942708333 0.947916667 0.953125 0.958333333 0.963541667 0.96875 0.973958333 0.979166667 0.984375 0.989583333 0.994791667 1 1 1 182 183 1 1 0.534 0.546 0.548 0.549 0.555 0.592 0.593 0.601 0.624 1 1 115 0.468 116 0.481 117 0.487 118 0.491 119 0.498 120 0.503 121 0.506 122 0.522 123 0.534 124 0.546 125 0.548 126 0.549 127 0.555 128 0.592 129 0.593 130 0.601 131 0.624 132 0.628 133 0.638 134 0.66 135 0.672 136 0.682 137 Grand Total 138 139 140 141 142 143 1 1 184 1 1 185 1 ܢܙ 1 1 186 187 1 1 1 ܢܙ 1 188 0.628 1 1 1 189 0.638 1 1 1 190 0.66 1 1 1 191 1 0.672 0.682 1 1 192 1 192 192
By observing data, 100% observations are less than 0.7.

And 56 observations lise in between 0.3 and 0.699.

c) All are observaion has ratio less than 0.7 and hence 100% of subjects had ratio less than 0.7.

56 observations are lies in between 0.3 and 0.6999 and hence 29.17% of subjects had these ratio.

I have attached the image in which all calculations are carried out. Frequency of each observation is obtain using pivot table such as on row all the unique observations and on column count of each unique.