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Question
In certain manufacturing process, it was found that the mean length of the parts produced by the lathe was 20.05 mm, with the standard deviation of 0.02 mm. The lower and the upper specification limits for the length are 20.03 mm and 20.08 mm respectively.

a) What proportion of part s will be out of the specification limit.

b)Find the probability that a part selected at random would have a length less than 20.01?

c)The life time of a certain type of motor is normally distributed with the mean of 10 years and the standard deviation of 2 years. If the manufacturer is willing to replace only 3% percent of the motor that fail how long  a guarantee should offer?

I was unable to get to the problem until now, but here at last is my answer.

Since the mean is 20.05 and the standard deviation is 0.02, how far away the limits are from the center can be found.  Since 20.05 - 20.03 = 0.02, the lower limit is 1 standard deviation below the mean.  Since 20.08 - 20.05 = 0.03, the upper limit is 1.5 standard deviations above the limit.

(a) To find the answer, find the probability that a normal variable is outside the range (-1, .5).

Take a normal distribution table, so I'll use
https://stat.duke.edu/~banks/111-FAQ.dir/StdNormCumDistTable.pdf

The value at -1 is 0.1587.
The value at 1.5 is one minus the value at -1.5, which is 1 - 0.0668 = 0.9332.

The chance the value is between -1 and 1.5 is 0.9332 - 0.1587 = 0.7745.

(b) Since 20.01 is 0.04 less than the average, which is 2 standard deviations away,
the probability of that is seen to be 0.0228.

(c) For this one, we have to find how many standard deviations give a chance of 0.03.
Reading the table, that is roughly at 1.88 standard deviations away from the normal.

Converting that back to the distribution we have means taking 10 - 2(1.88) = 10 - 3.76 = 6.24.
This is the same as roughly 6 years, 3 months.  So if it goes out before then, he should replace it.

Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Very Clear and Detailed Answer. Thank you very much.

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#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.