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Manufacturer of metal pistons finds that on averagc 12% of his pistons are defectivea) What is thc probability that batch of 6 pistons will contain defective piston...

Question

Manufacturer of metal pistons finds that on averagc 12% of his pistons are defectivea) What is thc probability that batch of 6 pistons will contain defective pistons? b) What is the probability that a batch of 6 pistons will contain at Icast defective pistons?c) What is the expected number of defective pistons in batch o 6? d) What is the standard deviation of the number of defective pistons in a batch of 6?

manufacturer of metal pistons finds that on averagc 12% of his pistons are defective a) What is thc probability that batch of 6 pistons will contain defective pistons? b) What is the probability that a batch of 6 pistons will contain at Icast defective pistons? c) What is the expected number of defective pistons in batch o 6? d) What is the standard deviation of the number of defective pistons in a batch of 6?



Answers

An automotive industry supplier produces pistons for several models of automobiles.
Twenty samples, each consisting of 200 pistons, were selected when the process was
known to be operating correctly. The numbers of defective pistons found in the samples
follow.
$$\begin{array}{cccccccccc}{8} & {10} & {6} & {4} & {5} & {7} & {8} & {12} & {8} & {15} \\ {14} & {10} & {10} & {7} & {5} & {8} & {6} & {10} & {4} & {8}\end{array}$$
$\begin{array}{l}{\text { a. What is an estimate of the proportion defective for the piston manufacturing process }} \\ {\text { when it is in control? }} \\ {\text { b. Construct the } p \text { chart for the manufacturing process, assuming each sample has } 200} \\ {\text { pistons. }}\end{array}$
$\begin{array}{l}{\text { d. Compute the upper and lower control limits for an } n p \text { chart }} \\ {\text { e. Answer part (c) using the results of part (d). }}\end{array}$

All right, so, the probability that all six work is are are able to replicate is going to be 36 -12 back. Can't divided by the 36 that there are to choose from. That's going to be the probability of the first one is able to replicate. And since these events are independent, we can multiply them. So the probability of the second one is going to be 36 -2. All 12, that can't minus the one that we took out from before, and then we're gonna continue this uh minus one fashion each time we go up one, so the next one will be minus two And we are going to go through six. So that will be 36 -12, that cans minus the five that we went through over 36 -5. And this is going to turn out to be a zero 069. And then for part B, what is the probability that at least one is not capable of replication? And the probability of at least one is the probability of all of them working, one minus all of them working, so all six And that is gonna be 1 -069, Which is equal to 0.931.

Hello everyone. We will be going to solve problems 1 94. This problem is given that there are 36 bacteria cells in a badge out of which do. All of the cells are not capable. We love. The cells are not capable of Salida replication. Not 24 but perfect. No, in the easy part Now six C, electorate random without replacement. Now we have two in the A part. We have to find the property that all six cells of this electric cells are able to replicate. So it will be 24 x 36 Into 23 x 35. As we are being without replacement. 22 x 34. 21 x 33, 25:32 And 19531. We just used a calculator to calculate this. This is coming out to be 0.069. This will be answered with a part. Now the people they're asking what is the probability that at least one of the selected cells is not capable of replication? This will just be the actions of probability will be one minus all the cells are capable of replication so that we have already calculated Should be for nine three fund. This will be honest for the B part. Thank you

So for this problem, we have three plants manufacturing hard drives and shipping them to a warehouse for distribution. Plant one produces 54% of inventory I for inventory and has 4% defect rate land to not pant plant to has or produces 35% of the inventory and has an 8% defect rate. Plant three produces the remainder, so that's 100 percent minus 54%. Minus 35% is equal to 11% of inventory, and it has a 12% the fact rate. So first thing that we want is we want to draw a tree diagram that represents this information. So we start off going to make sure I give myself a lot of space here. Okay, so we start off, then we could go plant one. I'm going to call that event a one. And that has a probability of 0.54 then. Yeah, once we have determined that we were from plant one, then we have the possibility that it is defective. All say that defective is event Be so we have probability of defective given it came from plant one. So that is defective. That is going to be 4% or 0.4 going to zoom in a little bit and I'm going to make my pan a little bit smaller here. So that is 0.4. Then we have not defective, going to say not defective his event be prime. And I'm just going to say Good gets us there. Alternatively, could come from Plant one Key A to which is equal to E. A to Waas. Where is it? 35% or 0.35 that is plant to. Similarly, we have probability of be given a to which is equal to Oh, I just realized I made a mistake. That was 0.4 there. That should be 0.4 and that probability of not defective up there should be 0.96 We have probability of effective given it came from Plant to is 0.8 which means then that's the probability that it's not defective, given it came from plant to equal to 0.92 then we have a probability came from Plant three is equal to 0.11 and the probability that that is defective probability of be given a three is equal to 0.12 and the probability it is not defective, given a came from Plant three is equal. Thio zero point eight eight. So that catalogs all the information that we were given for the next problem. We want to find the probability that a warehouse inspector select one hard drive at random and finds that it is both defective and from plant to So we can figure that out well, in in terms of interpreting that that is the event or were wanting to find the probability that event a two occurs and event be it occurs. So it was from plant to, and it is defective that we can get just thinking about the logic of going down our, um, tree diagram. It would be a 0.35% chance of going down this branch. So a 0.35% chance that it was from plant to and then given that there's a 0.8% chance that it is defective to arrive at the conclusion that both of those events have happened without it being conditional, we basically just multiply together the two probabilities that 0.35 on 0.8 Actually, that should be an ax to so make a little bit easier to read. And I'll note that, of course, that is the same thing as probability that be occurs, given a two occurs times the probability that a two occurs so 0.35 times 0.8 35 times 0.8 is going to be so it is a 0.28 or a 2.8% chance that it occurs than Part C. We are asked, what is the probability that a randomly selected hard drive is defective? The probability that a randomly select the hard drive is defective, that is, overall, that is the probability of event. Be happy here. We can use the law of total probability because the A events a one, a two and a three are mutually exclusive and exhaustive. So it's either from plant one plant to or plant three. That's the exhaustive and mutually exclusive. It can't be from both Plant one and from plant to. So the law of total Probability here tells us that the probability it's defective, equal to the probability that it is effective, given that it came from 81 times the probability that it came from a one plus probability of be given a to times probability of a to plus the probability of be given a three, given that it is from Plant three. So the probability of we can look back at what we have up here, so we had be given a one is 0.4 Yeah, your 0.4 times ability of a one is 0.54 plus probability of be given a to 0.8 times probability of a to was 0.35 plus probability of be given a three 0.12 time's up. That should have a P out front 0.12 times probability that it came from Plant three, which is 0.11 putting all of that together, throwing it into a calculator. Get that? The total probability of something being effective is 0.6 to 8 or 6.28%. Andi, for the last part, we suppose that a hard drive is defective. We want to determine what the probability is that it came from plant to So here we want the probability of a to given event be which is obviously going in a bit of a different direction. We know probability of be given a two, but not a two given b. So for this problem, we can apply Bayes theorem. So for our specific case, we could have that the probability of a two given B is equal to the probability of be given a to times the probability of a to divided by the sum from K or sorry from I equals one up to K, where K is the number of different possible events. So we have cables three. If the sum from I equals 123 the probability of be given a sub i times the probability of a I so that would give us the be given A to that was p of A to divided by be of be given a one on P of a one plus p of be given a to Times eight or probability of a to plus p of be given a three times probability of a three. I'm going to pause and throwing the numbers here so we would have on top 0.8 times 0.35 The bottom. We have 0.4 times, 0.54 plus 0.8 times, 0.35 plus 0.11 times 0.12 So the result of plugging all of that in I calculated it off screen. Result of calculus of playing all that in going to be 0.44 um six. Or you can approximate that to a 0.45 That is the probability that a defect if we have a defective product, then it came from plant to

This problem. We have a company that produces goblets and 10% of its goblets have cosmetic flaws. So for a were asked, if six a randomly selected how likely is it that only one has a cosmetic flaw? So this is the probability mass function for finding exactly one out of six the probability of success of 10% or 0.1. Remember, generally, that's written a number of successes, number of trials and the probability of success. And using a calculator or software, we get zero point 354 and for B were asked out of six randomly selected goblets. What is the probability that, at least to have cosmetic flaws, we're looking for a probability then X is greater than or equal to two. And we can also view this as one minus the probability that X is less than or equal to one, and we can write This. Azaz follows the cumulative distribution for one success, so that means getting zero or one successes out of six trials. Probability of success is 0.1, and this is equal to one minus zero point 88 57 If you use software to you evaluate a cumulative probability, and this equals 0.114 and then for part C or asked if the goblets air examined one by one. What is the probability that at most five must be selected to find four that do not have cosmetic defects? So two word that another way in the sample of five. We're looking for at least four that do not have cosmetic de defects. We must have at least four out of those five that do not have cosmetic defects, which means that we can have at most one out of a five that does have a cosmetic defect. And so and because having a cosmetic defect is a success in our example, this is the cumulative probability for getting one success out of five trials, where the probability of success is 0.1 and calculating that we get zero point 91 85


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