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Question Appendix.C.5: A normal distribution has a mean of 20 and a standard deviat...

A normal distribution has a mean of 20 and a standard deviation of 1.0. Find the probabilities:

a.           A value that is 22 or less.

b.           A value that is 22 or more.

App.C.5
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First, determine the value of z for 22:

z = \frac{x  –  \mu}{\sigma} = \frac{22  –  20}{1} = + 2.00

a. From Appendix B Table B, the area (probability) to the left of z = +2.00 is .9772.

Table B.1  Areas under the standardized normal curve, from −∞ to − z

.09 .08 .07 .06 .05 .04 .03 .02 .01 .00 z
.0002 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 −3.4
.0003 .0004 .0004 .0004 .0004 .0004 .0004 .0005 .0005 .0005 −3.3
.0005 .0005 .0005 .0006 .0006 .0006 .0006 .0006 .0007 .0007 −3.2
.0007 .0007 .0008 .0008 .0008 .0008 .0009 .0009 .0009 .0010 −3.1
.0010 .0010 .0011 .0011 .0011 .0012 .0012 .0013 .0013 .0013 −3.0
.0014 .0014 .0015 .0015 .0016 .0016 .0017 .0018 .0018 .0019 −2.9
.0019 .0020 .0021 .0021 .0022 .0023 .0023 .0024 .0025 .0026 −2.8
.0026 .0027 .0028 .0029 .0030 .0031 .0032 .0033 .0034 .0035 −2.7
.0036 .0037 .0038 .0039 .0040 .0041 .0043 .0044 .0045 .0047 −2.6
.0048 .0049 .0051 .0052 .0054 .0055 .0057 .0059 .0060 .0062 −2.5
.0064 .0066 .0068 .0069 .0071 .0073 .0075 .0078 .0080 .0082 −2.4
.0084 .0087 .0089 .0091 .0094 .0096 .0099 .0102 .0104 .0107 −2.3
.0110 .0113 .0116 .0119 .0122 .0125 .0129 .0132 .0136 .0139 −2.2
.0143 .0146 .0150 .0154 .0158 .0162 .0166 .0170 .0174 .0179 −2.1
.0183 .0188 .0192 .0197 .0202 .0207 .0212 .0217 .0222 .0228 −2.0
.0233 .0239 .0244 .0250 .0256 .0262 .0268 .0274 .0281 .0287 −1.9
.0294 .0301 .0307 .0314 .0322 .0329 .0336 .0344 .0351 .0359 −1.8
.0367 .0375 .0384 .0392 .0401 .0409 .0418 .0427 .0436 .0446 −1.7
.0455 .0465 .0475 .0485 .0495 .0505 .0516 .0526 .0537 .0548 −1.6
.0559 .0571 .0582 .0594 .0606 .0618 .0630 .0643 .0655 .0668 −1.5
.0681 .0694 .0708 .0721 .0735 .0749 .0764 .0778 .0793 .0808 −1.4
.0823 .0838 .0853 .0869 .0885 .0901 .0918 .0934 .0951 .0968 −1.3
.0985 .1003 .1020 .1038 .1056 .1075 .1093 .1112 .1131 .1151 −1.2
.1170 .1190 .1210 .1230 .1251 .1271 .1292 .1314 .1335 .1357 −1.1
.1379 .1401 .1423 .1446 .1469 .1492 .1515 .1539 .1562 .1587 −1.0
.1611 .1635 .1660 .1685 .1711 .1736 .1762 .1788 .1814 .1841 −0.9
.1867 .1894 .1922 .1949 .1977 .2005 .2033 .2061 .2090 .2119 −0.8
.2148 .2177 .2206 .2236 .2266 .2296 .2327 .2358 .2389 .2420 −0.7
.2451 .2483 .2514 .2546 .2578 .2611 .2643 .2676 .2709 .2743 −0.6
.2776 .2810 .2843 .2877 .2912 .2946 .2981 .3015 .3050 .3085 −0.5
.3121 .3156 .3192 .3228 .3264 .3300 .3336 .3372 .3409 .3446 −0.4
.3483 .3520 .3557 .3594 .3632 .3669 .3707 .3745 .3783 .3821 −0.3
.3859 .3897 .3936 .3974 .4013 .4052 .4090 .4129 .4168 .4207 −0.2
.4247 .4286 .4325 .4364 .4404 .4443 .4483 .4522 .4562 .4602 −0.1
.4641 .4681 .4721 .4761 .4801 .4840 .4880 .4920 .4960 .5000 −0.0

Table B.2  Areas under the standardized normal curve, from −∞ to + z

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7 .7580 .7611 .7642 .7673 .7703 .7734 .7764 .7794 .7823 .7852
.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997
3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

b. Because the total area under the curve is 100 percent, or 1.0000, the area to the right of z = +2.00 is simply 1.0000 − .9772 = .0228.

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