he mean of a normal probability distribution is 60; the standard deviation is 5.a. About what percent of the observations lie between 55 and 65?  b. About what percent of the observations lie between 50 and 70?  c. About what percent of the observations lie between 45 and 75? (Round your answers to 1 decimal place.)

Question
Asked Oct 23, 2019

he mean of a normal probability distribution is 60; the standard deviation is 5.

a. About what percent of the observations lie between 55 and 65?

 

 

b. About what percent of the observations lie between 50 and 70?

 

 

c. About what percent of the observations lie between 45 and 75? (Round your answers to 1 decimal place.)

 

 
check_circleExpert Solution
Step 1

Given mean of the normal distribution = 60
Standard deviation = 5

The Z score of 55 and 65 are calculated as shown below, we get Z scores as -1 and 1 respectively

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Step 2

With respect to below mentioned z tables P(Z ≤ -1) = 0.1587 and P( Z ≤ 1) = 0.8413
P(-1 ≤ Z ≤ 1) = P(Z ≤ 1) - P(Z ≤ -1) = 0.8413 - 0.1587 = 0.6826

The percentage of observations between 55 and 65 is 68.3 % (rounded to 1 decimal)

0.00
0.02
0.01
0.03
0.04
0.05
0.06
0.09
0.07
0.08
-3.0 0.0013
0.0013
0.0013
0.0012
0.0012
0.0011 0.0011 0.0011 0.0010 0.0010
0.0
0.02
0.05
0.01
0.03
0.04
0.06
0.07
0.09
0.08
-2.9 0.0019 0.0018
0.0017
0.0016 0.0015 0.0015 0.0014 0.0014
0.0018
0.0016
05080
0.5478
0.0
0.5000
0.5040
0.5120
0.5160
0.5199
0.5239
0.5279
0.5319
0.5359
0.0025 0.0024 0.0023
0.0023
0.0022 0.0021 0.0021 0.0020 0.0019
-2.8
0.0026
0.1
0.5398
0.5438
0.5517
0.5557
0.5596
0.5636
0.5675
0.5714
05753
-2.7 0.0035
0.0030 0.0029 0.0028 0.0027 0.0026
0.0034 0.0033 0.0032
0.0031
0.2
0.5793
0.5832
0.5871
0.5910
0.5948
0.5987
0.6064
0.6103
0.6141
0.6026
-2.6 0.0047 0.0045 0.0044 0.0043
0.0041
0.0040 0.0039 0.0038 0.0037 0.0036
0,6255
0.3
0.6179
0.6217
0.6293
0.6331
0.6368
0.6406
0.6443
0.6480
0.6517
-2.5 0.0062
0.0060
0.0055
0.0054 0.0052 0.0051
0.0049 0.0048
0.0059
0.0057
0.4
0.6844
0.6879
0.6554
0.6591
0.6628
0.6664
0.6700
0.6736
0.6772
0.6808
-2.4
0.0080
0.0073
0.0082
0.0078
0.0075
0.0071 0.0069 0.0068 0.0066 0.0064
0.5
0.6915
0.6950
0.6985
0.7019
0.7054
0.7088
07123
0.7157
0.7190
0.7224
-2.3
0.0107
0.0104 0.0102 0.0099 0.0096
0.0094 0.0091 0.0089 0.0087 0.0084
0.6
0.7257
0.7291
0.7324
0.7357
0.7389
0.7422
0.7454
0.7486
0.7517
0.7549
0.0129
-2.2
0.0139 0.0136
0.0132
0.0125
0.0122 0.0119 0.0116
0.0113 0.0110
0.7
0.7580
0.7611
07642
0.7673
0.7704
0.7734
0.7764
0.7794
0.7823
0.7852
0.8023
-2.1
0.7881
0.7910
0.7939
0.7995
0.8051
0.8078
0.8106
0.8133
0.0158 0.0154
0.0150
0.0143
0.8
0.7967
0.0179 0.0174
0.0170
0.0166
0.0162
0.0146
0.9
0.8289
0.8340
0.8389
0.8159
0.8186
0.8212
0.8238
0.8264
0.8315
0.8365
-2.0 0.0228 0.0222
0.0217
0.0202
0.0188
0.0212
0.0207
0.0197
0.0192
0.0183
1.0 08413 0.8438
1.1 0.8643
0.8461
0.8485
0.8508
0.8531
0.8554
0.8577
0.8599
0.8621
-1.9 0.0287
0.0274
0.0268
0.0244
0.0239
0.0281
0.0262
0.0256
0.0250
0.0233
0.8686
0.8729
0.8665
0.8708
0.8749
0.8770
0.8790
0.8810
0.8830
-1.8 0.0359
0.0344
0.0301
0.0351
0.0336
0.0329
0.0322
0.0314
0.0307
0.0294
1.2
0.8849
0.8869
0.8888
0.8907
0.8925
0.8944
0.8962
0.8980 0.8997
0.9015
-1.7
0.0446 0.0436
0.0427
0.0418
0.0409
0.0401 0.0392
0.0384
0.0375 0.0367
-1.6
0.0516 0.0505
0.0548
0.0537
0.0526
0.0495 0.0485 0.0475 0.0465 00455
-1.5 0.0668
0.0606 0.0594 0.0582 0.0571 0.0559
0.0655
0.0643
0.0630
0.0618
-1.4
0.0749
0.0808
0.0793
0.0778
0.0764
0.0735 0.0721 0.0708 0.0694 0.0681
0.0951
0.0838 0.0823
-1.3
0.0968
0.0934
0.0918
0.0901
0.0885
0.0869 0.0853
0.1131
0.1093
-1.2
0.1151
0.1112
0.1075
0.1056 0.1038 0.1020 01003 0.0985
-1.1 0.1357 0.1335
0.1271
0.1251
0.1210
0.1314
0.1292
0.1230
0.1190 0.1170
-1.0 0.1587 0.1562
0.1469
01423
0.1401
0.1539 0.1515
0.1492
0.1446
0.1379
-0.9 0.1841
0.1635
0.1611
0.1814
0.1788
0.1762
0.1736
0.1711
0.1685
0.1660
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Image Transcriptionclose

0.00 0.02 0.01 0.03 0.04 0.05 0.06 0.09 0.07 0.08 -3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 0.0 0.02 0.05 0.01 0.03 0.04 0.06 0.07 0.09 0.08 -2.9 0.0019 0.0018 0.0017 0.0016 0.0015 0.0015 0.0014 0.0014 0.0018 0.0016 05080 0.5478 0.0 0.5000 0.5040 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 -2.8 0.0026 0.1 0.5398 0.5438 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 05753 -2.7 0.0035 0.0030 0.0029 0.0028 0.0027 0.0026 0.0034 0.0033 0.0032 0.0031 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6064 0.6103 0.6141 0.6026 -2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 0,6255 0.3 0.6179 0.6217 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 -2.5 0.0062 0.0060 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 0.0059 0.0057 0.4 0.6844 0.6879 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 -2.4 0.0080 0.0073 0.0082 0.0078 0.0075 0.0071 0.0069 0.0068 0.0066 0.0064 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 07123 0.7157 0.7190 0.7224 -2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.0129 -2.2 0.0139 0.0136 0.0132 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 0.7 0.7580 0.7611 07642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8023 -2.1 0.7881 0.7910 0.7939 0.7995 0.8051 0.8078 0.8106 0.8133 0.0158 0.0154 0.0150 0.0143 0.8 0.7967 0.0179 0.0174 0.0170 0.0166 0.0162 0.0146 0.9 0.8289 0.8340 0.8389 0.8159 0.8186 0.8212 0.8238 0.8264 0.8315 0.8365 -2.0 0.0228 0.0222 0.0217 0.0202 0.0188 0.0212 0.0207 0.0197 0.0192 0.0183 1.0 08413 0.8438 1.1 0.8643 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 -1.9 0.0287 0.0274 0.0268 0.0244 0.0239 0.0281 0.0262 0.0256 0.0250 0.0233 0.8686 0.8729 0.8665 0.8708 0.8749 0.8770 0.8790 0.8810 0.8830 -1.8 0.0359 0.0344 0.0301 0.0351 0.0336 0.0329 0.0322 0.0314 0.0307 0.0294 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 -1.6 0.0516 0.0505 0.0548 0.0537 0.0526 0.0495 0.0485 0.0475 0.0465 00455 -1.5 0.0668 0.0606 0.0594 0.0582 0.0571 0.0559 0.0655 0.0643 0.0630 0.0618 -1.4 0.0749 0.0808 0.0793 0.0778 0.0764 0.0735 0.0721 0.0708 0.0694 0.0681 0.0951 0.0838 0.0823 -1.3 0.0968 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.1131 0.1093 -1.2 0.1151 0.1112 0.1075 0.1056 0.1038 0.1020 01003 0.0985 -1.1 0.1357 0.1335 0.1271 0.1251 0.1210 0.1314 0.1292 0.1230 0.1190 0.1170 -1.0 0.1587 0.1562 0.1469 01423 0.1401 0.1539 0.1515 0.1492 0.1446 0.1379 -0.9 0.1841 0.1635 0.1611 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660

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Step 3

The Z score of 50 and 70 are calculated as shown below,...

X-
Z =
For X 50
50-60
Z =
5
-10
-2
5
For X 70
70-60
Z =
5
10
-=2
5
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Image Transcriptionclose

X- Z = For X 50 50-60 Z = 5 -10 -2 5 For X 70 70-60 Z = 5 10 -=2 5

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