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What cutoff value bounds the lowest 30% when using a normal model with a mean of 40 and a standard deviation of 4? What cutoff values bond the middle 71% when using a normal model with a mean of 52 and a standard deviation of 9?

Question

What cutoff value bounds the lowest 30% when using a normal model with a mean of 40 and a standard deviation of 4?

 

What cutoff values bond the middle 71% when using a normal model with a mean of 52 and a standard deviation of 9?

check_circleAnswer
Step 1

Given normal model with mean 40 and standard deviation 4.
With reference to Standard normal table the Z score such that are to its left is 0.3015 is -0.52 and 0.2981 area to the left of Z score -0.53. We can say approximately the Z score for area of 0.30 to its left is average os -0.52 and -0.53, which is -0.525.

0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
3.0
0.0013
0.0012
0.0012
0.0011
0.0010
0.0013
0.0013
0.0011
0.0011
0.0010
-2.9
0.0018
0.0017
0.0019
0.0018
0.0016
0.0016
0.0015
0.0015
0.0014
0.0014
-2.8
0.0023
0.0021
0.0026
0.0025
0.0024
0.0023
0.0022
0.0021
0.0020
0.0019
-2.7
0.0035
0.0034
0.0033
0.0032
0.0031
0.0030
0.0029
0.0028
0.0027
0.0026
0.0047
0.0044
0.0043
0.0041
0.0038
0.0037
-2.6
0.0045
0.0040
0.0039
0.0036
-2.5
0.0062
0.0060
0.0059
0.0057
0.0055
0.0054
0.0052
0.0051
0.0049
0.0048
-2.4
0.0082
0.0080
0.0078
0.0075
0.0073
0.0071
0.0069
0,0068
0.0066
0.0064
0.0107
0.0104
0.0102
0.0099
0.0091
0.0089
-2.3
0.0096
0.0094
0.0087
0.0084
0.0113
-2.2
0.0139
0.0136
0.0132
0.0129
0.0125
0.0122
0.0119
0.0116
0.0110
0.0162
0.0179
0.0174
0.0166
0.0154
0.0150
0.0146
0.0143
-2.1
0.0170
0.0158
-2.0
0.0228
0.0222
0.0217
0.0212
0.0207
0.0202
0.0197
0.0192
0.0188
0.0183
-1.9
0.0287
0.0281
0.0274
0.0268
0.0262
0.0256
0.0250
0.0244
0.0239
0.0233
1.8
0.0351
0.0307
0.0359
0.0344
0.0336
0.0329
0.0322
0.0314
0.0301
0.0294
-1.7
0.0446
0.0436
0.0427
0.0418
0.0409
0.0401
0.0392
0.0384
0.0375
0.0367
1.6
0.0548
0.0537
0.0526
0.0516
0.0505
0.0495
0.0485
0.0475
0.0465
0.0455
-1.5
0.0668
0.0643
0.0618
0.0655
0.0630
0.0606
0.0594
0.0582
0.0571
0.0559
-1.4
0.0808
0.0793
0.0778
0.0764
0.0749
0.0735
0.0721
0.0708
0.0694
0.0681
-1.3
0.0968
0.0918
0.0885
0.0869
0.0853
0.0951
0.0934
0.0901
0.0838
0.0823
-1.2
0.1151
0.1131
0.1112
0.1093
0.1075
0.1056
0.1038
0.1020
0.1003
0.0985
1.1
0.1357
0.1335
0.1314
0.1292
0.1271
0.1251
0.1230
0.1210
0.1190
0.1170
-1.0
0.1587
0.1515
0.1446
0.1423
0.1562
0.1539
0.1492
0.1469
0.1401
0.1379
-0.9
0.1762
0.1841
0.1814
0.1788
0.1736
0.1711
0.1685
0.1660
0.1635
0.1611
-0.8
0.2119
0.2033
0.1977
0.1949
0.1922
0.1894
0.1867
0.2090
0.2061
0.2005
-0.7
0.2420
0.2327
0.2389
0.2358
0.2296
0.2266
0.2236
0.2206
0.2177
0.2148
-0.6
0.2743
0.2709
0.2676
0.2643
0.2611
0.2578
0.2546
0.2514
0.2483
0.2451
f0.2981
0.3336
-0.5
0.3015
0.2912
0.3085
0.3050
0.2946
0.2877
0.2843
0.2810
0.2776
0.3372
-0.4
0.3409
0.3446
0.3300
0.3264
0.3228
0.3192
0.3156
0.3121
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Image Transcriptionclose

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 3.0 0.0013 0.0012 0.0012 0.0011 0.0010 0.0013 0.0013 0.0011 0.0011 0.0010 -2.9 0.0018 0.0017 0.0019 0.0018 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 -2.8 0.0023 0.0021 0.0026 0.0025 0.0024 0.0023 0.0022 0.0021 0.0020 0.0019 -2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 0.0047 0.0044 0.0043 0.0041 0.0038 0.0037 -2.6 0.0045 0.0040 0.0039 0.0036 -2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 -2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0,0068 0.0066 0.0064 0.0107 0.0104 0.0102 0.0099 0.0091 0.0089 -2.3 0.0096 0.0094 0.0087 0.0084 0.0113 -2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0110 0.0162 0.0179 0.0174 0.0166 0.0154 0.0150 0.0146 0.0143 -2.1 0.0170 0.0158 -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 -1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 1.8 0.0351 0.0307 0.0359 0.0344 0.0336 0.0329 0.0322 0.0314 0.0301 0.0294 -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -1.5 0.0668 0.0643 0.0618 0.0655 0.0630 0.0606 0.0594 0.0582 0.0571 0.0559 -1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.3 0.0968 0.0918 0.0885 0.0869 0.0853 0.0951 0.0934 0.0901 0.0838 0.0823 -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 -1.0 0.1587 0.1515 0.1446 0.1423 0.1562 0.1539 0.1492 0.1469 0.1401 0.1379 -0.9 0.1762 0.1841 0.1814 0.1788 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 -0.8 0.2119 0.2033 0.1977 0.1949 0.1922 0.1894 0.1867 0.2090 0.2061 0.2005 -0.7 0.2420 0.2327 0.2389 0.2358 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 -0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 f0.2981 0.3336 -0.5 0.3015 0.2912 0.3085 0.3050 0.2946 0.2877 0.2843 0.2810 0.2776 0.3372 -0.4 0.3409 0.3446 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121

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Step 2

Using the below mentioned formula of Z score we determine the value of X which is cutoff value which bounds lowest 30%.
The cut off value, which bounds lowest 30% is 37.9

X-
Z =
-0.525 = -40
4
X- 402.1
X 37.9
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X- Z = -0.525 = -40 4 X- 402.1 X 37.9

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Step 3

Given normal distribution with mean 52 and standard deviation 9.
middle 71% of the normal distribution has 29% on both tails, where each tail equally divided with 14.5% as shown below

With reference to Z table the Z score with 0.14...

0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
-3.0
0.0012
0.0011
0.0013
0.0013
0.0013
0.0012
0.0011
0.0011
0.0010
0.0010
-2.9
0.0017
0.0015
0.0015
0.0019
0.0018
0.0018
0.0016
0.0016
0.0014
0.0014
-2.8
0.0022
0.0019
0.0026
0.0025
0.0024
0.0023
0.0023
0.0021
0.0021
0.0020
71%
-2.7
0.0034
0.0035
0.0033
0.0032
0.0031
0.0030
0.0029
0.0028
0.0027
0.0026
0.0044
-2.6
0.0047
0.0045
0.0043
0.0041
0.0040
0.0039
0.0038
0.0037
0.0036
14.5%
0.0057
0.0055
0.0049
-2.5
0.0062
0.0060
0.0059
0.0054 0.0052
0.0051
0.0048
14.5%
0.0080
0.0075
0.0071
-2.4
0.0082
0.0078
0.0073
0.0069
0.0068
0.0066
0.0064
0.0102
-2.3
0.0107
0.0104
0.0099
0.0096
0.0094
0.0091
0.0089
0.0087
0.0084
0.0132
0.0125
0.0122
-2.2
0.0139
0.0136
0.0129
0.0119
0.0116
0.0113
0.0110
0.0179
0.0166
0.0162
0.0150
-2.1
0.0174
0.0170
0.0158
0.0154
0.0146
0.0143
0.0228
0.0207
0.0192
-2.0
0.0222
0.0217
0.0212
0.0202
0.0197
0.0188
0.0183
-1.9
0.0250
0.0244
0.0287
0.0281
0.0274
0.0268
0.0262
0.0256
0.0239
0.0233
1.8
0.0344
0.0329
0.0294
0.0359
0.0351
0.0336
0.0322
0.0314
0.0307
0.0301
1.7
0.0436
0.0446
0.0427
0.0418
0.0409
0.0401
0.0392
0.0384
0.0375
0.0367
-1.6
0.0475
0.0548
0.0537
0.0526
0.0516
0.0505
0.0495
0.0485
0.0465
0.0455
1.5
0.0668
0.0655
0.0643
0.0630
0.0618
0.0606
0.0594
0.0582
0.0571
0.0559
-1.4
0.0808
0.0721
0.0793
0.0778
0.0764
0.0749
0.0735
0.0708
0.0694
0.0681
1.3
0.0968
0.0951
0.0934
0.0918
0.0901
0.0885
0.0869
0.0853
0.0838
0.0823
-1.2
0.1151
0.1131
0.1112
0.1093
0.1075
0.1056
0.1038
0.1020
0.1003
0.0985
-1.1
0.1292
0.1271
0.1251
0.1357
0.1335
0.1314
0.1230
0.1210
0.1190
0.1170
0.1446
-1.0
0.1423
0.1587
0.1562
0.1539
0.1515
0.1492
0.1469
0.1401
0.1379
-0.9
0.1841
0.1814
0.1788
0.1762
0.1736
0.1711
0.1685
0.1660
0.1635
0.1611
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Image Transcriptionclose

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 -3.0 0.0012 0.0011 0.0013 0.0013 0.0013 0.0012 0.0011 0.0011 0.0010 0.0010 -2.9 0.0017 0.0015 0.0015 0.0019 0.0018 0.0018 0.0016 0.0016 0.0014 0.0014 -2.8 0.0022 0.0019 0.0026 0.0025 0.0024 0.0023 0.0023 0.0021 0.0021 0.0020 71% -2.7 0.0034 0.0035 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 0.0044 -2.6 0.0047 0.0045 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 14.5% 0.0057 0.0055 0.0049 -2.5 0.0062 0.0060 0.0059 0.0054 0.0052 0.0051 0.0048 14.5% 0.0080 0.0075 0.0071 -2.4 0.0082 0.0078 0.0073 0.0069 0.0068 0.0066 0.0064 0.0102 -2.3 0.0107 0.0104 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 0.0132 0.0125 0.0122 -2.2 0.0139 0.0136 0.0129 0.0119 0.0116 0.0113 0.0110 0.0179 0.0166 0.0162 0.0150 -2.1 0.0174 0.0170 0.0158 0.0154 0.0146 0.0143 0.0228 0.0207 0.0192 -2.0 0.0222 0.0217 0.0212 0.0202 0.0197 0.0188 0.0183 -1.9 0.0250 0.0244 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0239 0.0233 1.8 0.0344 0.0329 0.0294 0.0359 0.0351 0.0336 0.0322 0.0314 0.0307 0.0301 1.7 0.0436 0.0446 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 -1.6 0.0475 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0465 0.0455 1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 -1.4 0.0808 0.0721 0.0793 0.0778 0.0764 0.0749 0.0735 0.0708 0.0694 0.0681 1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.1 0.1292 0.1271 0.1251 0.1357 0.1335 0.1314 0.1230 0.1210 0.1190 0.1170 0.1446 -1.0 0.1423 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1401 0.1379 -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611

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