An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean

hexacordoK 2020-11-20 Answered
An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean of 116 cm and a standard deviation of 4.8 cm.
A. Find the probability that one selected subcomponent is longer than 118 cm.
B. Find the probability that if 4 subcomponents are randomly selected, their mean length exceeds 118 cm.
C. Find the probability that if 4 are randomly selected, all 4 have lengths that exceed 118 cm.
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Expert Answer

opsadnojD
Answered 2020-11-21 Author has 95 answers

Step 1
Given Data:
XN(116,4.82)
A) To Find: P(X>118)
P(X>118)=P(Xμσ>118μσ)
=P(z>1181164.8)
=P(z>0.4167)
=1P(z<0.4167)
=10.662=0.338 (using s tan dard normal distribution table)
The probability that one selected subcomponent is longer than 118 cm is 0.338
Step 2
B) sample size=4
Therefore,XN(116,4.824)
To Find:P(X>118)
P(X>118)=P(Xμσ/n>118μσ/n)
=P(z>1181164.8/4)
=P(z>0.8333)
=1P(z<0.8333)
=10.798=0.202 (using standard normal distrbution table)
The probability that if 4 subcomponents are randomly selected, their mean length exceeds 118 cm is 0.202
Step 3 C) P(all 4 lengths exceed 118 cm)=P(X>118)4
=0.3384 (From part A, P(X>118)=0.338)=0.0131
The probability that if 4 are randomly selected, all 4 have lengths that exceed 118 cm is 0.0131

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