Question

The systolic blood pressures of the patients at a hospital are normally distributed with a mean of 136 mm Hg and a standard deviation of 13.8 mm Hg. Find the two blood pressures having these properties: the mean is midway between them and 90% of all blood pressures are between them.

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Answer 1

The two values are calculated by finding the Z scores at the corresponding hence

The bell curve of the normal distribution is 100% probability since the curve is symmetrical hence the middle 90 % of are leaves 5% in the left which can be said as lower 5% and 5% area in the upper end which can be called as higher 5% area. and the Z scores at the respective points is calculated using Z table shown below.

we will find that

Z at a lower 5% of the normal curve is calculated as -1.65 which is computed using Z table hence by Z formula

and at the top 5%, the Z value is 1.65 hence by Z

So, the middle 90 % lie between {113.23, 158.77}

Z table

X-136 X- 1.65 Z X = 113.23 13.8

X-136 X-H1.65 Z X 158.77 13.8

Table of Standard Normal Probabilities for Negative Z-scores Table of Standard Normal Probabilities for Positive Z-scores 0.00 0.5000 05398 0.03 0.5120 0.04 0.09 0.5359 0,00 0,06 0.5239 0.02 0.08 001 003 D.04 0,05 0.06 0.07 0,08 0.09 001 0.02 0,05 0.07 7 0,0002 0,0003 .0004 0.0006 0,0003 00 0.5040 0.5080 0.5199 0.5279 D.5675 0.6064 0.5319 -34 0,0003 0.0003 0.0003 0.0003 0.0003 00003 0.0003 0.5160 0.0005 0.0007 0.5438 0.5753 -3.3 0.0005 0.0005 0,0004 0,0004 00004 0.0004 0.0O04 0.0005 0.0007 0.0010 0.0014 D.D003 0.1 0.5478 0.5517 0.5557 0.5596 D5636 0.5714 -3.2 0.0007 0.0006 0.0009 0.0006 0,0006 00006 0.0005 0,0005 0,2 0.5793 0.5832 0.5871 0,5910 0.5948 0.587 0.6026 0.6103 0,6141 -3.1 0.0010 0.6517 0,6879 0.7224 D.DO09 a.0009 0.0008 0.0008 0.0008 00008 D.DO07 D.0010 0.0014 0.3 0.6179 0.6217 0.6255 0.6293 0,633 0.6368 D.6406 D.6443 0.6480 0,6554 0.6915 O0011 0.0015 06591 0,6950 -3.0 0.0013 0.D013 0.0013 0,0018 .0012 0.0012 0.0016 0.0011 0.0011 0.4 0.6628 0.6664 0.7019 0.6700 0.6736 0.7088 06772 0.7123 D.6808 0.6844 -2.9 0.0019 0,0018 0.0017 0.0016 0.0015 0,5 0,6985 07054 0.7157 0.7190 0.7517 D.0025 0.0034 0.0045 O7389 0.7704 0.7995 D.7486 0.7794 0.8078 DS340 D.8577 D8790 2.8 0.0026 0.0024 a.0023 0,0023 0.0031 0,0041 0.0022 0.0030 0,0040 O0021 0.0021 0.0020 D.DO19 0.6 0.7257 0.729 0.7324 0.7357 07422 D.7454 0.7549 0.7852 0.8133 0.7823 0.8106 -2.7 -2.6 0.0035 0 0017 0.0033 0,0044 0.0032 0.0029 00039 0.0028 0.0038 0.005 0.0027 0.0026 0.0036 0.7 0.8 0.7580 0.7881 0.7611 0.7910 0.7642 0.7673 0,7967 0.7734 0.8023 0.7764 0.805 0.0043 0.0037 0.7930 1 08389 0.8621 0.8159 0.8413 -25 0.0062 D.DO60 0.0059 a.0057 0.0055 0.0073 0.0054 00052 0.0049 D.D048 0.9 0.8186 0.8212 0,8238 .8264 0.8289 D8315 0.8365 0.8599 0.8810 0.0080 -24 0.0082 0,0078 0.0102 0.0075 0.0071 00059 0.0068 0.0066 0.0064 1.0 0.8438 a5665 0.8869 0.9049 0,8461 0.8485 0.8508 0.8531 0.3749 0.8554 03770 0.0107 0.0139 0.0179 0.009 0.8643 23 0.0104 a.0099 0.0096 0,0004 0.0122 0.0158 0.0089 0.0087 D.0O84 0.8686 0.8708 O.8725 0.8830 0.0136 0.0174 -2.2 -2.1 0.0132 0.0170 0.0129 0,0166 0,0125 0.0162 0.0110 0.0143 1.2 1.3 0.8907 0,9082 O.8925 0.8944 0.9115 0.8997 0,9015 00119 0.0116 0.0113 0.8849 0.8888 0.8962 0.913 09279 0.8980 00154 0.0150 0.0146 0.9032 0.9066 0.9147 0.9162 0.9177 09099 0,9222 0,9357 0,9474 0.9573 0.9656 0.9207 0,9236 0.9370 -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 00250 00314 0.0192 0.0188 0.0183 1.4 0.9192 0.9332 09251 0.9265 D.9292 0.9306 0.9319 0.9429 0.9535 0.9625 0.0287 0.0359 0.0256 0.0239 0.0233 1.5 0.9382 D.9418 D.9525 0,9441 -1.9 0.0281 0.0274 0,0268 0,0262 0.0244 0.9345 0.9394 0.9505 0.9599 0.9406 0.00 0.0329 0.0294 D.067 -L.8 0.035 0.0436 0.0344 0.0336 0.0322 0.007 1.6 0,9452 0.9554 0.9463 0,9484 09495 0.9515 0,9545 0.040 00495 0.0375 0,0465 0.057 0.9582 -1.7 0.0446 0.0427 0,0418 0,0409 00392 0.0384 1.7 0.9564 09591 0.9608 0.9616 0,9633 0048s 00594 0.0455 0.0559 09686 0.9750 0.0548 0.0526 0.0516 0.0475 0,9649 0.9706 16 0.0537 00505 1.8 0.9641 0,9664 09671 0.9678 0.9693 0.9699 0.9761 0.9812 0.9854 0.0668 0.0655 0,0643 0.0630 0.0618 0,0606 0.0582 1,9 09713 0,9719 0.9726 0,9732 09738 09744 0.9756 0,9767 -1.5 O0721 00869 0.1038 0.0708 0.0853 0.0808 D.U793 U.U778 14 .0764 .0749 0.0735 0.0694 D.0681 2,0 0,9772 0.9778 0.9783 0.9788 0.9793 09798 0.9803 D.98U8 0.9817 0.0968 0.1151 01357 0.1587 0.184 0.9834 0,987 -1.3 0.0918 0.0823 0.9826 09838 0.9857 0.0951 0.0934 0.0901 0.0885 0.0838 2.1 0.9821 0.9830 0.9842 0,9878 0.9846 D.9850 0.1020 0.1210 0988 0,9864 0,9868 -1.2 0.1131 0.1112 0,1093 0.1075 0.1056 0.125 0.1003 D.0985 2.2 0.9861 0.9875 0.9884 0.9887 0,9890 0.9901 0.9925 0.9906 0.9929 -1.1 0.1335 01314 .1292 0.1271 0.1492 O1230 0.1190 D.1170 23 0.9893 0,9806 0.9898 09904 0.9927 0.9909 0.9911 0.9913 0,9916 0.140 0993 0.9922 -10 0.1562 0.1539 0.1515 0.1469 0.1446 0.1423 0.1379 2.4 0.9918 0.9920 0.9940 09935 0.9932 0.9934 0.995 0.9936 0.1814 0.9946 -09 0.1788 a.1762 .1736 0.1711 0.1977 0.2266 O1685 0.1660 D.1635 D.1611 D.1867 2.5 0.9938 0.9941 0,9943 09945 0.9948 0.996 0.9971 D.9949 0.9952 0.2061 0.2358 0.9957 0,9968 0.9953 0.9965 D.9962 O.8 0.2119 0.2090 0.2033 0.2327 0.2005 0.2296 O.1949 0.1922 0.1894 2.6 0,9956 09959 0.9960 0.9963 0.9964 0.9967 0.9972 09973 D.9979 0.9985 0.9989 -07 0.2420 0.2389 02236 0.2206 0.2177 0.2148 D.245 0.2776 0.3121 2.7 0,9966 09969 0.9970 0,9974 0.2578 0.2912 0.3264 06 0.2743 0.2709 0.2676 d.2643 0.2611 02546 02514 0.2483 2.8 0.9974 0.9975 0.9976 0.9977 09977 0.9978 D.9979 0.9980 0.998 0.9984 0.9989 0.9981 0.9987 0,9982 0,9987 0.9985 0.9989 -05 -04 0.3085 0.34-16 0.3050 0.3409 D.3783 0.3015 0.3372 0,2981 0,3336 0.2946 0.3300 02877 03228 0.2843 0.3192 0.2810 0.3156 2,9 3.0 0.9982 0.9987 0,999 0.9983 0,9988 09984 0,9988 0.9986 0.9990 0.9986 0,9990 03 0.3821 0.3745 0.4129 g.3707 0.3669 0,4052 0.3632 03594 0.3557 0.3936 04325 0.3520 D.3483 3.1 0.9990 0.9991 0.9993 0.9991 09992 0.9992 09992 D.9992 0.9993 0.9993 0.9991 0,9995 -02 0.4207 04602 0.4168 0,4090 0,4013 0.3974 0.3897 0.3859 3.2 0.9993 0,9994 0.9994 09994 0.9994 09996 0.9995 0.9995 0,9995 0.4562 0.4522 0.4483 04443 04404 04364 0.4286 04247 3.3 .9995 0,9995 0.9996 09996 0.9996 D.9996 0.9996 0.9997 0480 0476 D468 0.4840 0,9997 D.9997 0.500 0.4960 0 4920 0.4880 0.4721 D4641 3.4 0.9997 0,9997 0,9997 09997 0.9997 09997 0.9997 0.9998 Note that the probabilities given in this table represent the area to the LEFT of the z-score. The area to the RIGHT of a z-score 1- the area to the LEFT of the z-score