Question

The defect length of a corrosion defect in a pressurized steel pipe is normally distributed with mean value 28 mm and standard deviation 7.2 mm. (a) What the probability that defect length is at most 20 mm? Less than 20 mm? (Round your answers to four decimal places.) at mast 20mm less than 20mm (b) What is the 75th percentile of the defect length distribution-that is, the value that separates the smallest 75% of all lengths from the largest 25%? (Round your answer to three decimal places.) מרח (c) What is the 15th percentile of the defect length distribution? (Round your answer to three decimal places.) mm three decimal places.) (d) What values separate the middle 80% of the defect length distribution from the smallest 10% and the largest 10%? (Round your answers smallest 10% mm largest 10% mm You may need to use the appropriate table in the Appendix of Tables to answer this question. Need Help? Talk to a Tutor

Answer 1

The solution is done showing all steps with calculations and required explanations. For your better understanding, a z table is also attached. Hope you can understand and appreciate the work.

X Let Z = X-M Let x denotes the defect length of a corrosion effect. N(Mean = M =28, standard. Deviation - 7.2) x-M NCO,) le PCZ +2) = E(2) & 06-Z) = 1-&(z) [EC.) values from z - table (a) PC & £20)=P(x L 20-28) = P(Z <=1-11) [(-1.11) = 1 – (111) -1-o.8665 = 011335 (Auswer) • PCX < 20) = (-1.11) = 0.1335 (Answer) (6) Let la' be the 75th percentile. P(x ha) or, P(X-M-<a - 28 ) = 0.75 By 672) -0.75 - (0.675) 0.75 os a-28 7-2 3.675 Oy a = 28+ 7. 2x0.675 = 32.86 :.75th percentile is 32.86 mm. is 32.86 mm. (Answer) Let (e) cy? be the 15th percentile P(X< b) =or15 :) = 0.15 on P(x a 6-28) Į (6-28 ) = 0.15 = 0.15 E &(-1.035) oy b-28 7.2 =-11035 o ba 28.-1.0 35x7.2 = 20,548 ,. 15th (Auswah 201548 percentile is we)

be (c' mm (d) Smallest 10% PCX <s) - 0.10 O, PCXML625 ) - 0.10 G = 0.10 = (1.28) c-28 oly 7.2 -) e-28 7:2 -- 128 9, c - 28-1:28x7.2 - 18.784 .. Smallest 10% is 18.784 mm. (Auswen) be ďmm Largest 10% PCX <d)=0.90 or, P(X <d-28) =0.90 By $(?)=0.903 -0.90 = $(1.28) Q d = 28+7.2 x 1.28 - 37.216 37.216 mm. (Auswen) 10% is Largest

z table :

Z 0.0 0.1 02 03 0.4 05 0.6 0.7 0.8 0.9 1.0 1.1 12 13 14 15 1.6 17 18 1.9 2.0 21 22 23 2.4 25 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 0.00 0.5000 05398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 09192 09332 09452 09554 0.9641 0.9713 0.9772 0.9821 0.986] 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.998] 0.9987 0.9990 0.9993 0.9995 0.9997 0.01 0.5040 0.5438 0.5832 0.6217 | 0.6591 10.6950 10.7291 0.7611 0.7910 0.8186 0.8438 0.8665 10.8869 | 0.9049 0.9207 0.9345 0.9463 09564 0.9649 0.9719 0.9778 10.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 | 0.9975 10.9982 | 0.9987 0.9991 0.9993 0.9995 0.9997 0.02 0.03 0.5080 0.5120 0.5478 0.5517 0.5871 0.5910 0.6255 0.6293 0.6628 0.6664 0.6985 0.7019 0.7324 0.7357 07642 0.7673 0.7939 0.7967 0.8212 0.8238 0.8461 0.8485 0.8686 0.8708 0.8888 0.8907 0.9066 0.9082 0.9222 0.9236 0.9357 0.9370 0.9474 0.9484 0.9573 0.9582 0.9656 0.9664 0.9726 0.9732 0.9783 0.9788 0.9830 0.9834 0.9868 0.9871 0.9898 0.9901 0.9922 0.9925 0.9941 0.943 0.9956 0.9957 0.9967 0.9968 0.9976 0.9977 0.9982 0.9983 0.9987 0.9988 0.9991 0.9991 0.9994 0.9994 0.9995 0.9996 0.9997 0.9997 0.04 05160 0.5557 05948 0.6.33] 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 09495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.9992 0.9994 0.9996 0.9997 0.05 05199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 10.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989 0.9992 0.9994 0.9906 0.9997 0.06 0.5239 0.5636 0.6026 0.6406 10.6772 07123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 10.9131 10.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 10.9881 10.9909 0.9931 0.9948 0.9961 0.9971 10.9979 0.9985 10.9989 0.9992 0.9994 0.9996 0.9997 0.07 0.5279 0.5675 0.6064 0.6443 0.6808 07157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989 0.9992 0.9995 0.9996 0.9997 008 05319 0.5714 0.6103 0.6480 0.6844 07190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 09162 0.9306 09429 09535 0.9625 0.9699 0.9761 0.9812 09854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990 0.9993 0.9995 0.9996 0.9997 0.00 05359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.88.30 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.99 16 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998