Question

HIV/AIDS patients are regularly monitored for their CD4 counts in order to make sure the antiretroviral therapies are effective. Suppose the distribution of CD4 counts in a population is approximately normal with mean = 237 and standard deviation = 43. If HIV patients are defined as progressing to AIDS after their CD4 counts are less than 200, what proportion of this population is considered to be suffering from AIDS?

Answer 1

Given that the population of CD4 count is normally distributed so, Z score can be applicable for proportion calculation,

Given that the mean $\mu=$ 237 and Population standard deviation as $\sigma$ = 43, now if the CD4 count is less than 200 then there will be HIV.

To find the proportion below X =200 we need to find the Z score as:

-u Z = 200 – 237 -= -0.88 43

so, P(X<200) = P(Z<0.88) can be calculated using excel formula for normal distribution which is =NORM.S.DIST(-0.88, TRUE) which results in probability = 0.1894.

so, the proportion that can be considered suffering from AIDS is 0.1894.

The proportion can be also calculated using the Z table shown below:

Table of Standard Normal Probabilities for Negative Z-scores Table of Standard Normal Probabilities for Positive Z-scores 2 00 0.1 -3.2 30 -2.7 0.6 0.7 0.8 0.9 1.0 0.00 001 0.02 0.03 0.04 0.05 0.06 0.07 0 0. .09 -34 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0. 0 3 0.0002 -3.3 0.0005 0.0005 0.000 0.0004 0.0004 OCKX140 X140 XX4 0.024 0.0003 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.035 0.0005 0.0010 0.0009 0.0009 (1.0009 0.0008 0.0IS OCCIS D IS 0.007 0.0007 0.0013 D.DOI3 0.0013 0.0012 0.0012 0.00110 0011 0.00110 .0 10 D.0010 -29 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 2014 D.2014 28 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.002 0.0021 0 0 20 D.0019 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 00029 0.0028 0.027 0.0026 0.0047 0,0015 0,0014 0.0013 0.001 0.0010 00039 0.0038 0 37 0.0036 25 0.0062 D.DOX0 0.0059 0 .0057 0.0055 0.0054 0.0052 0.00 0.0.49 0.0048 -2.4 0.0082 0.0080 0.0078 0.0075 0.00730.0071 00069 0.0068 0 66 0.0064 0.0107 D.DID4 0.0102 CV) 0006 0.0004 0.000 0.00 0.087 D.0084 -2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0 0119 0.0116 0. 13 0.0110 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 00154 0.0150 00 46 0.0143 0.0008 0.0272 0.0217 0.0212 0.0207 00202 00197 0.0192 0. 0 8 0.0183 -1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.02SD 0.0244 0.039 0.0233 -1.8 0.0359 0.035 0.0344 0916001200002200114 0.047 001 D.0204 -1.7 0.1446 0.0436 0.0427 0.0418 0.0409 .0401 00192 0.0194 00 75 0.0367 0.0548 0.0537 0.0826 0.0S16 0.OSOS 0.0495 00485 0.0475 0.0165 0.0435 -1.5 0.7068 017655 0.0643 0.06) 0.0618 0.0606 00594 0.0592 0.0171 0.0559 -1.4 0. 06 0.0793 0.077801.0764 0.0749 0.0735 00721 0.0708 0094 0.0681 -1.3 0.0968 0.0951 0.09340,0918 0.0001 0.088500046900853 0.038 0.0823 0.1151 0.111101112 01001 0.1075 0.1056 0.1018 01020 0. 13 0.0085 -1. 1 0 1357 0.1335 0.13140 .1292 0.1271 0.1251 0.1230 0.1210 0.11 90 0.1170 -10 0.1587 0.1562 0.1549 01515 0.1492 0.1469 0.1446 0.1423 01 01 0.1379 0.1841 01314 0.1788 0.1762 0.1 736 0.17110 .1685 0.1660 K 0.1611 -0.8 LA A M AD A ANA 0.1894 0.1867 07 0.2420 0.2389 0.2358 0 .2327 0,2296 0,2266 0.2236 0.2206 0.2148 0.2743 0.2709 0.2676 0.2643 0.2611 0.2575 0.2946 0.2514 0.2483 0.2451 05 0.3085 0.3050 0.3015 0 .2981 0,2916 0.2912 0.2877 0.2813 0.2810 0.2776 04 0.34.16 0.3409 0.3372 0.3336 0.3300 (3264 0.3228 03192 0.3156 0.3121 0.3821 0 .3783 0.3745 0.3702 0.3669 0.3632 3594 0.3557 0.3520 D.3483 -0.2 0.4207 0.4168 0.4129 0 .1000) 0.4052 0.4013 0.3974 0.3936 0.2897 0.3859 01 0.16020.4562 0.4522 .4483 0.4443 0.4404 0.4364 04325 0.4286 0.4247 -0.0 0.50m) 0.496) 0.4920 (1.4880 0.4840 0.4801 0 4761 0.4721 0.46810.4641 0.00 0 01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.00 0.5000 5040 0,5080 0.5120 05160 05199 05239 0.5279 0.5319 05359 0.5398 0.5438 0.5478 0.5517 0.5557 1.5596 0.5636 D 5675 0.5714 0.5753 0.5793 0.5832 0.58710.5910 059.18 0.5987 0.6026 0.6061 06103 0.6141 0.6179 0.6217 0.6255 0.6793 0.6331 0.6468 06406 0.6443 0.6480 0.6517 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6806 0.6844 0.6879 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7580 0.7611 0.7642 0.7673 07704 0.7734 0.7764 0.7794 0.7823 0.7852 0.7881 0.7910 0.7939 0.7967 0.7995 0.8022 0.8051 0.8078 0.8106 0.8133 0.8159 0.816 0.8212 0.82330 .3264 0.329 0.3315 D.5240 0.8365 0.8359 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 0.8643 2.8665 0.8686 0.875 0.8720 08740 0.8770 D.870 0.8810 0.8890 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 01.9147 0.9162 0.9177 0.002 0.9207 0.9222 0.9236 09251 09265 09279 0.92020 .9306 0.9319 0.9332 0.9345 0.9357 0.9320 019382 0.9394 0.9406 0.9418 0.9429 0.9441 0.0452 0.0463 0.0474 0.0484 00495 09505 09515 0.9525 0.9535 0,9545 0.0554 0.9564 0.0473 09582 009910 9590 DO .8616 0.0625 0.9633 0.9641 0.9649 0.9656 0.9664 0.9621 0.9678 0.9686 0.9693 0.9699 0.9706 0.9713 0.9719 0,9726 0,9732 0,0738 0.9744 09750 9756 0.9761 09767 0.9772 0.9778 0.97830.9788 09793 0.9798 0.98030.9808 0.9812 0.9817 0.9821 0.9826 0.0830 0.98 34 09838 0.9842 0.9846 0.9850 0.9854 0.9857 0.9861 09864 09868 0.9871 09875 0.9878 0.988 D 9884 09887 0.980N 0.9893 0.9896 0.9898 0.9901 0.9904 0.99.06 0.99x09 0.9911 0 .9913 0.9916 0.9918 0.9920) 0.9922 0.9925 09927 9929 0.9931 0.9932 0.9934 0.9936 0.9938 0.9940) 0.0941 0.943 0 9945 0.9946 0.9948 D. 9949 0.99s1 0.0952 0.9953 0.99ss 0.99/56 0.957 09959 0.9960 0.9961 0.9962 0.9963 0.9964 09965 0.9966 09967 0.9268 00069 0.9970 0.9971 0.9972 09973 0.9974 0.9974 0.9975 0.9976 0.99770.9977 0.9978 0.9979 0.9979 0.99800.9981 0.9981 0.9982 0.9982 0.0983 09984 0.9984 0.9985 0.9985 0.9986 0.9986 09987 0.998709987 0.9988 09988 0.9989 0.99890998909990 0.9990 0.999200.9991 0.9991 0.991 09992 0. 9 2 0.9992 D.9992 0.9993 0.9993 0.9993 0.9993 0.9991 0.9994 09994 0.9991 0.9994 0.9995 0.9995 0.9995 0.9995 0.6995 0.0095 0.96 09095 0966 0.9906 0.9996 0.9996 0.9997 0.9997 0.9997 0.9997 0.997 090297 0X170907 09997 0.9997 0.9993 2.0 2.1 2.6 3.0 -03 3.4 Note that the probabilities given in this table represent the area to the LEFT of the 7-score. The area to the RIGHT of a z-score = 1- the area to the LEFT of the Z-score