Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 5.2millimeters (mm) and a standard deviation of 1.4 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)(a) the thickness is less than 3.0 mm (b) the thickness is more than 7.0 mm (c) the thickness is between 3.0 mm and 7.0 mm

Question
Asked Jul 16, 2019

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 5.2millimeters (mm) and a standard deviation of 1.4 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)

(a) the thickness is less than 3.0 mm
 

(b) the thickness is more than 7.0 mm
 

(c) the thickness is between 3.0 mm and 7.0 mm
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Step 1

Given Thickness measurement of ancient prehistoric Native american pot shards discovered in hopi village are approximately normally distributed.

Mean is 5.2 mm and standard deviation is 1.4 mm.

To calculate the probability that thickness less than 3 mm P(X < 3), where X is thickness and it is normally distributed. 

We find the Z score for X = 3 as shown below, we get Z score is -1.57.

So we can write P( X < 3) = P( Z < -1.57)

X
Z=4-u
σ
3-5.2
Z =
1.4
Z =-1.57
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X Z=4-u σ 3-5.2 Z = 1.4 Z =-1.57

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Step 2

With reference to the below shown Z table P(Z < -1.57) = 0.0582. So we can say that P(X < 3) = 0.0582.

The probability that thickness less than 3 mm is 0.0582

0.06
0.07
0.08
0.09
0.00
0.01
0.02
0.03
0.04
0.05
0.0013
0.0012
0.0011
0.0010
-3.0
0.0013
0.0013
0.0012.
0.0011
0.0011
0.0010
0.0016
-2.9
0.0019
0.0018
0.0018
0.0017
0.0016
0.0015
0.0015
0.0014
0.0014
-2.8
0.0024
0.0023
0.0023
0.0021
0.0020
0.0019
0.0026
0.0025
0.0022 0.0021
-2.7
0.0033
0.0031
0.0026
0.0035 0.0034
0.0032
0.0030
0.0029 0.0028
0.0027
0.0044
0.0040
0.0036
0.0047
0.0045
0.0043
0.0041
0.0039
0.0038
0.0037
-2.6
0.0048
-2.5
0.0062
0.0060
0.0059
0.0057 0.0055
0.0054
0.0052
0.0051
0.0049
0.0073
0.0082
0.0080
0.0078
0.0075
0.0071
0.0069
0.0068
0.0066
0.0064
-2.4
0.0099
0.0084
-2.3
0.0107
0.0104
0.0102
0.0096
0.0094
0.0091
0.0089
0.0087
0.0132
0.0125
0.0113
0.0110
-2.2
0.0139
0.0136
0.0129
0.0122 0.0119
0.0116
0.0170
0.0166
0.0162
0.0158
0.0143
-2.1
0.0179
0.0174
0.0154
0.0150
0.0146
0.0212
0.0207
0.0202
0.0183
-2.0
0.0228
0.0222
0.0217
0.0197
0.0192
0.0188
0.0274
-1.9
0.0287
0.0281
0.0268
0.0262
0.0256
0.0250
0.0244
0.0239
0.0233
0.0329
-1.8
0.0359
0.0351
0.0344
0.0336
0.0322
0.0314
0.0307
0.0301
0.0294
0.0427
0.0409
0.0375
0.0367
0.0446
0.0436
0.0418
0.0401
0.0392
0.0384
-1.7
0.0526
0.0516
0.0505
0.0495
0.0455
-1.6
0.0548
0.0537
0.0485
0.0475
0.0465
0.0594 C0582 0.0571
0.0643
-1.5
0.0668
0.0655
0.0630
0.0618
0.0606
0.0559
0.0735
-1.4
0.0808
0.0793
0.0778
0.0764
0.0749
0.0721
0.0708
0.0694
0.0681
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0.06 0.07 0.08 0.09 0.00 0.01 0.02 0.03 0.04 0.05 0.0013 0.0012 0.0011 0.0010 -3.0 0.0013 0.0013 0.0012. 0.0011 0.0011 0.0010 0.0016 -2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0015 0.0015 0.0014 0.0014 -2.8 0.0024 0.0023 0.0023 0.0021 0.0020 0.0019 0.0026 0.0025 0.0022 0.0021 -2.7 0.0033 0.0031 0.0026 0.0035 0.0034 0.0032 0.0030 0.0029 0.0028 0.0027 0.0044 0.0040 0.0036 0.0047 0.0045 0.0043 0.0041 0.0039 0.0038 0.0037 -2.6 0.0048 -2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0073 0.0082 0.0080 0.0078 0.0075 0.0071 0.0069 0.0068 0.0066 0.0064 -2.4 0.0099 0.0084 -2.3 0.0107 0.0104 0.0102 0.0096 0.0094 0.0091 0.0089 0.0087 0.0132 0.0125 0.0113 0.0110 -2.2 0.0139 0.0136 0.0129 0.0122 0.0119 0.0116 0.0170 0.0166 0.0162 0.0158 0.0143 -2.1 0.0179 0.0174 0.0154 0.0150 0.0146 0.0212 0.0207 0.0202 0.0183 -2.0 0.0228 0.0222 0.0217 0.0197 0.0192 0.0188 0.0274 -1.9 0.0287 0.0281 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 0.0329 -1.8 0.0359 0.0351 0.0344 0.0336 0.0322 0.0314 0.0307 0.0301 0.0294 0.0427 0.0409 0.0375 0.0367 0.0446 0.0436 0.0418 0.0401 0.0392 0.0384 -1.7 0.0526 0.0516 0.0505 0.0495 0.0455 -1.6 0.0548 0.0537 0.0485 0.0475 0.0465 0.0594 C0582 0.0571 0.0643 -1.5 0.0668 0.0655 0.0630 0.0618 0.0606 0.0559 0.0735 -1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0721 0.0708 0.0694 0.0681

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Step 3

To calculate the probability that thickness greater than 7 mm P(X > 7), where X is thickness and it is normally distributed. 

...

Z =X-u
7-5.2
Z =
14
Z =1.29
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Z =X-u 7-5.2 Z = 14 Z =1.29

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