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Question 3 (6 marks) XYZ Manufacturing has 6 machines that perform & particular task Breakdowns occur frequently for this machine. Past records indicate that th...

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Question 3 (6 marks) XYZ Manufacturing has 6 machines that perform & particular task Breakdowns occur frequently for this machine. Past records indicate that the number of; bceakdowns that occur each day is described by the following probability distribution:Number of Breakdowns Probability 0.4More than 3What is the expected number of breakdowns in any given day? b. What is the variance for this distribution? What is the probability that there will be at least 2 breakdowns in a day?

Question 3 (6 marks) XYZ Manufacturing has 6 machines that perform & particular task Breakdowns occur frequently for this machine. Past records indicate that the number of; bceakdowns that occur each day is described by the following probability distribution: Number of Breakdowns Probability 0.4 More than 3 What is the expected number of breakdowns in any given day? b. What is the variance for this distribution? What is the probability that there will be at least 2 breakdowns in a day?



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A factory manager collected data on the number of equipment breakdowns per day. From those data, she derived the probability distribution shown in the following table, where $W$ denotes the number of breakdowns on a given day. $$\begin{array}{c|ccc} \hline \boldsymbol{w} & 0 & 1 & 2 \\ \hline \boldsymbol{P}(\boldsymbol{W}=\boldsymbol{w}) & 0.80 & 0.15 & 0.05 \\ \hline \end{array}$$ a. Determine $\mu_{W}$ and $\sigma_{W}$ b. On average, how many breakdowns occur per day? c. About how many breakdowns are expected during a 1 -year period, assuming 250 work days per year?

Right. So this question involves a person process for a mechanic shop. The average is 1.5 breakdowns each day. And the question asked, What's the probability that there at least three breakdowns in a given day? So that's the same thing is asking one minus. The probability is, we don't what so p of zero plus p of one and p of two and notice. Since we're something the probabilities here it's just one minus the cumulative from ability. So Paulson CDF in your G I t floor and the mean is 1.5 and we're summing up to two, and that works out to be 0.1912

In this question, we have to find the expected value of X. What we have with us is a table. First of all, what is the formula for expected value? The formula for expected value of X is nothing but submission off X p o fix submission off XP FX. We have different values effects. If we look at the table, we have number of repair calls as eggs. It will be 012 and three and then we have respective probabilities off 30.1 point 3.4 and point to. Okay, So what is going to be my expected value summation off. So let's just write it. This is going to be zero into 0.1 right, Plus one into 0.3 plus two in 2.43 pointers to in 2.4 plus three in 20.2. What does this become? This turns out to be 1.7, right. So this is 1.7 calls for our 1.7 calls for our alright. Moving on. How do you find obedience? The variance off X is going to be expected. Value off X minus meal square. Right. If I'm not wrong. This is the formula expected value off X minus mu square or one very handy formula is also this is equal toe expected value off X square minus mu square. We can use either of these. So let's try to do with one thing. Let's try to use the first one. What will this become? This will become zero minus 1.7 square in 20.1 plus one minus 1.7 square in 20.3 Full US. Tu minus 1.7 square into 0.4 plus three minus 1.7 square in 20.2. Right. This is going to be the variance. Let me call it V. So if I solve this, the various that I'm getting is 0.67167 So let me just write. This is 0.67 Okay. What is going to be standard deviation? It is going to be Sigma is going to be route off variance. Right. This is going to be somewhere around. If I take my calculator. This is square root off 0.67 which is 0.8185 0.8 185 calls 0.8185 calls per hour for our These are my answers

Welcome to new Madrid. In this current problem, we are observing the distribution of the diameters of bearings produced by a machining operation. That means that there is a machine. Right. And that's producing buildings 1234 A number of infinite number of items are being produced by the machining operation. So we have to visualize that whenever there is a normal distribution, the number of uh items in the sample would be very large. Okay, So now In the problem itself, it's given that new is 3.000 interest, right? And Sigma is equal to 0.0010". Okay. No, it is also mentioned that a particular bearing measurement okay bearing will be accepted only if that measurement of that particular beer Bearing falls within the integral 3.000 plus or minus 0.00 20. That means what if I have a normally distributed well, bearings being pretty uh introduction then say for example, I'm taking three. Okay? And then this is her plus 3.002. And this will be negative so 3 -0.002 within this bracket. Okay. If it falls then we exit and we reject if it falls outside this now we want to know how much of the total production gets wasted. Also we are willing to know the percentage in terms of percentage and also in terms of the probability. So let us try to find it out. So we are trying to find how much is being scrapped. So first start from there. Okay so how much? Yes being script. Okay, when you write statements like this, that gives you a better idea. So when X will not fall under this, correct? So reliability of X not belonging to this Interval 3.000 plus or -0.0020. Okay. that can be written as X not falling under or not coming within this interval 2.998. That's what we obtained when we subtract this from this And 3.002 when we have these two we get this right. So X should either be less than this value or greater than this value tonight. So let us strike foot pain. The ability X less than 2.998 plus X Greater than 3.002. Now this is where clearly we will she used uh normal probabilities. So now here we will take help of the standardization technique. Also, I know what I have written three points raises rates three point there's no five. I'm sorry about that. No, this is 3.0 to minus meal by sigma. The next thing we use ourselves so that we can make use of the tray table. Right So z listen 2.998 -3.005 divided by little point 001 Okay. Plus probability that greater than 3.00 to -3.005. You divided by 0.001. Now that would give probability of then listen -0.0025. Okay, so Well right negative of zero 0025 divided by 0.001 plus probability that greater than 0.0015 divided by 0.00. Which will it's simply huh? Mhm. And said Lisbon -2.5 plus pre viability that Greater than 1.5. Okay now let me get the normal changing over here. Yes. So now we will try to understand this from this numbers. Okay, so let this be -2.5 correct? And this is 1.5. So whatever comes under this, this entire region is being accepted, correct. And whatever is this is getting rejected, Right? So now from the tables we can directly obtain this value. But how will we obtain this value? So if you see whatever is below -2.5 is above plus 2.5. Right? So we can write probability Zed greater than 2.5. So now it's pretty simple. We just have to obtain the values from the table. Right? So let us bring the table now. So now if we see 1.5 for 1.5, the value is little point 0668. Uh huh. 0.06 68. Less For this, you have to do .5. So if you see 2.5 it's here. It's 0.062. So we were right, 0.0062. And if we add them we get 0.0730. So if this is the probability Then what is the percentage the percentage would be equivalent to multiplying it with 100%. Right? So we then get 7.3%. That is 7.3% of the total production process. Boardroom production process is script. Now if we want an applet probability, what would it be be? Will simply compare with regular table. So if this is the sad accident, score their access then we can think this is that excuse the original values. So even then it will be the same. It will between 1005 and you didn't have the readings over here. The the SD right. The SD is 0.001. So that means this should be this point should be 3.006. This should be 3.007, correct. And this should be 3.8 And the same with this, this this these three points can be obtained here and end up the day these two will be same. So we will have the same. Probably the 0.730 is the applet probability as well because Z and X are linearly related. That is X is equals two mu. That is 3.005 plus Sigma's it? That means 01001 into think so, I hope you could understand the explanation. Let me know if you have any questions. Bye bye.

Welcome to New Married. So in the current problem we are given that there is a machining operate. The operation will not which keeps on uh which keeps on producing the bearings with measurements, uh that follows a normal distribution. So, X will be the measurement of diameters. Right? So we will write and I will be in even dial meters in inches of buildings of hearings produced by the machining operations. Now, it is also mentioned that X for you normal distribution, that means new to be two point judo, you know zero. All right. You inches and the standard deviation to be mm hmm 0.10 Ahh. Alright. So what will be the probability that we will ask first? We understand these bearings are not all conflict. Some of them are good, some of them are bad. So, the bad ones that gets scrapped or you know, they get for before re machining. Those will be the only acceptance area for X will be that I intervened 3.0 plus minus treat. Zero point zero zero two. Okay. Which means X. If it belongs to this region 3.0 miners, 0.2 and 3.0 plus 0.2 Even accept. That is if if it becomes strong two 0.998 two, 3.0 two. So this is where we accept all the bearings. Now. They're asking that what percentage or what probability? Using whatever approach will be accepted and because whatever approach Beaty Beat athletes or be the appendix three tables, it will always be the seat. Now we didn't try to have uh when we try to visualize how are normal table. So if you see the mean is at 3.5 That means in the center of the data we must have 3.5 Then we know the next point will be at noon plus sigma. So this point should be this value plus the sigma. Well that means 3.1 Fight. The next point will be at new plus two sigma. That will be 3.25 The next point will be new plus three sigma. That it. So that will be the 3.35 So how would that normal table? Look, let's have a view. I have already kept the dragon ready for you. So let's so if you look at this current diagram, you can see that the center is that 3.0 05 Start it over here then 2.152 point 252.35 And on the same not the left side. And so now, what is our acceptance region? Our acceptance region is 2.998 to 3.0 to let us first right there. So our acceptance region is 2.998 two, 3.0 two. So first of all, what would be that region? So let us try to have a different end and 2.998 would be somewhere around it. I draw a lie even here and then we have 3.2 which would be something over. Yeah. So if I put another line, we're here, it would be something like this. Right? Mhm like this. So the area in between should be accepted by us. This entire area. This whatever the balls are produced within this area will be accepted. That means most of the production will be accepted by the production process and only a few. These are not acceptable is this idea. Listen this So now let us see how we can find that out. So first of all we will write we will not accept a uh bold if it is outside that range. Right? So we can write X not belonging to that 2.998 from a 3.2 In fact that means either it should be less than this value or greater than this. Well correct. So therefore probability of X less than 2.998 Okay or Mhm Great of them. 3.0 two. Now if we standardize what would happen, probability of decide. Also it's minus meal by sigma. Less than 2.998 minus new by sigma plus. Same with the side also. X minus mu by sigma. Greater than 3.0 to minus New by sigma. No, this can be written for that to be zero. Less than 2.998 minus 3.5 All divided by 0.1 plus. So the ability said greater than 3.2 minus 3.0 05 divided by 0.0 What? Here? We can find it simplify and right probability that less than minus 0.25 divided by 0.1 plus probability of zed greater than 0.0 15 divided by 0.1 Okay, no that simplifies to zero less than minus 2.5 plus. Is it later than 1.6. Now to understand this, I will again bring the normal table and explain how that looks. So if you see rapidly Tales yes then 3.6 so minus 2.5 would fly over. Yeah but if it is two point point minus two point not that whatever. Less than minus 2.5. He's also later than plus two points. So definitely we can change and write it to be probability the better than qualifiers. Plus the durability. Said they don't at one point. Right okay now we will bring the normal people. Okay the normal table with me beating us. The values of problems of that better than be points higher than 100. Right? So let us plan to find out now observed, observe over here. What is the values for 1.5 and 2.6? 1.5. This is the value for two point value. 2.5. This is the girl that means this is 0.62 plus 0.668 So if we add we get 0.730 This is the probability. And if we wanted the person did it will be 0.730 in 200. That is 7.3%. So 7.3% of the total people population. Total production is getting strapped. So do you understand what that means? That means that this stroll up and check. Yes. Yeah. So now we uh that that is actually 7.3%. Okay, so let us try to look at that. So if we see this extreme in the areas will construct up 0.730 point 073 or 7.3% of the total population. So I hope you could understand, let me know if you have any questions.


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