use-the-following-data-the-lifetimes-of-a-certain-type-of-automobile-tire-have-been-found-to-be-d-18

Question

Use the following data. The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of $100,000 \mathrm{km}$ and a standard deviation of $10,000 \mathrm{km}$ Answer the following questions.
What percent of the tires will last between $95,000 \mathrm{km}$ and $115,000 \mathrm{km} ?$

Barsha Rana · Accepted Answer

Welcome to numerous in the current problem. We are considering the lifetimes of tigers, Which is distributed normally between 10 to 10, uh relax kilometer and sorry, one like kilometer and sigma to be 10,000 kilometers. Now we are asked the question that what percentage of tires will last between 95,000 km, 2, 1, these many kilometers. So for that the first thing that we will start is funded rising. So we will have 95 triple zero minus 1, 2, 3, 4, 5, less than x minus late, Less than 115 000 -100000 divided by 10,000. Now this after being simplified, it will become minus this, correct? Because 100 minus 95 is five and 33 zeros and then less than this will be zed and this will be 15,000 divided by 10,000, which will give -0.5, less than zero, Less than 1.5. That is if there is a Standard normal distribution like this centered at zero. So this is something -0.5 and this is something say one. So this will be 1.5. So we want to know what percent will be this. So what we will do, we will take the entire area left to this, that is this, and then take the entire area left to this. And then we will find that probability Some probability said less than 1.5 Minour probability of zero, less than -0.5, Which is equivalent to probabilities said less than 1.5. This stays common but this becomes zero, greater than 0.5. Due to symmetry of normal distribution, which will be probability zed less than 1.5 -1 minus probability that less than 0.5. This is because the regular table, Yes, the regular table table that we have this will generally be giving the area less than if you can get a table that gives the value more than this. And you can directly that evaluate this. Now, going back to the values probabilities hit less than 1.5. years over here. So this is the value we are considering 0.9332 -1 plus. Probably to sit less than 0.5. So which is this? So it is 0.6915 6915. So how much will this value? B let us calculate. So For that I will simply use Excel. So I have a point line 33. Cool .9 metric -1 plus .6915. Eric. So we get 01 62 oh Little .6247. So we have 0.6247. Or if we do a percentage, it will be 62.47%. Right? So 62.47 percent. People will sort of tires Okay. This percent of tires will have a longevity of this.

Kayla Laughman · Answer

Okay, So we&#x27;re gonna look at how to evaluate our standard deviation and are mean in a normal distribution on. Then look at different percentages based off of that to see how much of our data would lie within the state or deviation that were given. So, for example, for this problem, let&#x27;s look at a certain life span of a tire. So if you buy a tire from a dealership before you have to get it replaced, it might have a certain amount of miles that you can have on it. So let&#x27;s just say the lifetime of this particular one is 30,000 miles, right? So we want to see that. So that would be our mean, which also come here in, like this. You and then our senior deviation, which is given to us, is 25 100 miles. So standard deviation. So typically, the average lifespan of this Tyree 30,000. But the standard deviation how far away from that? So they&#x27;re giving this interview aviation of 2500. So within 2500 of the mean would be the standard deviation. And that can also you&#x27;re in as such, Justin TV Asia. Okay, so We want to look at the percentage of tires that would have a mileage or the lifetime That would be between 20 2500 miles and 37,500 miles. So what percentage is going to be between those mileage numbers, huh? So to evaluate this, we would have to take and find dizzy that you and how we do that is we take X minus mu or are mean, if you will, over our standard deviation. Huh? That would be the formula you would use. And this is on Lee used for normal distribution. So if you remember normal distribution, that bell shaped curve where you mean, um, you would be right here, and then your standard deviations would go out this way. 12 and three, Negative. 12 and three. Standard deviations. Right. So that&#x27;s a normal bell curve. So we&#x27;re trying to find how many would fall within the 559,500 of the standard deviation, which they gave us of 2500 so soft we would simply plug in. The 1st 1 they&#x27;re giving us is the 22,500 which would be your axe, minus your mean, which was given, which is the 30,000 and divide by that standard deviation which again given to you. I think that hunter. Now, when you solve for that, you should get negative 3.0. So that would be your first Z that we would find negative three. Okay, that would be like yours, E. But we also have to find between 37,500. So the same exact thing we plug. So the acts that they&#x27;re giving us 37 1005 100 this time, Myers, the same given of your me over the standard deviation which you get given to you. And when she saw for this, you should get positive. Three. Okay, so that would be your second Z. Okay, so what does that mean? So now that we have positive three and a negative three as your Z&#x27;s on a normal curve, we would look to see what that would mean. Now, if you remember from the empirical rule within three, standard deviations of your mean is that 99.7%. And that&#x27;s just if you know it off your top of your head. So 123 standard deviations out a negative three. So your data that falls in between here. So how maney mileage is would fall within. That would be 99.7% for the empirical rule. Now, let&#x27;s say you weren&#x27;t sure how to find up, or you just couldn&#x27;t remember off the top of your head in your book. Or you can simply go online and search for what&#x27;s called a Z table, and what you do is you would find your values for three and negative three and you&#x27;ll see what it looks like on a curve. And it would basically give you the decimals so it would be 0.49 Eat seven if you really want to get granular with it on the point for 987 So that would be for each side. So if we were to draw this so that 0.4987 would just be one side, so maybe the negative and then the other side that would come together. So that&#x27;s why you have to add them so you would get the 99.74%. If you want to be that precise Okay, so that&#x27;s how you would find it in two different ways. So to go back to the problem, the percentage of having those mileage is of that tire between 559,500 would be 99.7% of the time would fall within that. Okay, Now, let&#x27;s say for if it&#x27;s not a normal distribution so it wouldn&#x27;t look like that bell curve and there was no additional information that you had about it being normally distributed, you could use a different theory. And that would state that, um, 89% of your data would be within three standard deviations of the mean so practically 89% 89% If it&#x27;s not a normal distribution, okay. And that&#x27;s within three standard deviations. So again, that was the easy that we had solved for, and you can use it off of a different the&#x27;re, um, heavy chose right here. Okay, here. So again, that&#x27;s if it&#x27;s not. If you didn&#x27;t get any additional information, that rule would just state within three standard deviations of your mean approximately 89% under that theory would fall within that. So that percentage between 559,500

Tyler Gaona · Answer

the lifetime of a tire is modeled on a normal distribution, with the mean lifetime being 22,000 miles and the standard deviation being 4000 miles and were asked two things first rest out of 4000 tires. How many tires do we expect to last more than 18,000 miles? So what we&#x27;re asking for is the probability that X is greater than 18,000. And again, with the normal distribution, you have to use a computer or look it up in a table, and you should get roughly 0.841 So that tells us that about 84% of the tires will last longer than 18,000 miles, so 84% of 4000 is about 300 3365 tires. So that&#x27;s about how many we can expect will last longer than 18,000 miles. And then we&#x27;re asked, What is the minimum number of miles we can expect that the tires will last for for 90% of the tires. So what we&#x27;re asking for here is something slightly different. Here were have something like number of miles m the probability, but that the tire last for longer than em, miles. We want that to be 90%. Okay? And now the issue here is we want to solve for em and so right, We can&#x27;t necessarily do this exactly, But we can at least get a good estimate again either by. So what you can do, for instance, is so we see 18,000 miles gives us roughly 84% chance. So next thing you might try Well, 17,000 miles, there should be more. Uh, we&#x27;re tires that will last for greater than 17,000 miles. Because we&#x27;re getting further away from the mean. You can try that and you can basically keep by trial and error. You confined value for M, which gets you pretty close to nine out of 10. And so if you experiment with a few values, I think you might find that&#x27;s 16,850 miles. Gives you a good estimate. There

Charles Carter · Answer

Okay, So, uh, first thing I do for this problem and set up a no hypothesis, so they know how about this would mean that there&#x27;s basically it&#x27;s not exciting. So that, uh, the basically said that whatever the entire company claims is true, So we can assume that the main is greater than or equal to the 50,000 mile mark. Okay, So just assuming that the claim is true, Which means the alternative hypothesis is that the main is less than 50,000. Okay, Yes, that we know that the data should be normally distributed in. Ah, um, write it the way the book, right. It writes it. Okay, so we have that means should be normally distributed around 50,000 with the standard deviation in this case I gave us to us is 8000 was the population. And we&#x27;re testing, uh, 28 tires says I am married 28. Okay, case the next. Put these values in from the problem. It was the test. Ah, the P value that I get a little like of the eight. I am getting a 0.0 for nine, which is close. But that just means there&#x27;s a um, there&#x27;s a chance of a type one error, which means rejecting. Ah, a correct no hypothesis, um, is 4.9%. So, um, because people use small, it&#x27;s gonna support our alternative hypothesis. Um, and since we&#x27;re testing according to an Alfa value of back 0.5 we&#x27;re looking for if it&#x27;s under 5% um, we would say that this data is, um is highly inconsistent. Okay, so we would say yes. This is highly and can consistent because, uh, RPI value is less than r calif of value current. Um, yeah. Okay. And I believe that&#x27;s all we need to say. All right. Thank you very much. I just like the main thing you want to make sure knows how to calculate this p value. I&#x27;m so calcula tion. 80 threes and 80 four&#x27;s haven&#x27;t under statistics suggest your Z test and pull again. The values I give in to you. Okay. Very much.

Neel Faucher · Answer

Okay, So in this problem were asked to find the probability that the vehicle is driven more than 15 dozen miles. We can read that probability X greater 15 1000. We know that from the problem that mean is equal to 12,494 94 on the standard deviation is equal to 1002 190. Now, given these pieces of information and the fact that this data is normally distributed, you can use the formula X minus mu over. Signal is you got to see. So we&#x27;re just gonna go ahead and do that for this number like that in its X 15,000 I was 12,490 for over. This is over 1290 is equal to 1.94 on. If you look at our see table that has a correspondent, probability is your point nine seven 38 Okay, now we&#x27;re going to be right this in terms of dizzy value, because that&#x27;s what we have here. And then input the probability to get our final answer. So figure I didn&#x27;t do that. We get probability. Let&#x27;s see. Was created 1.94 is equal to one minus a property. Z is less than 1.94 which is equal to one minus 0.97 three A. Just go to is your appoints your 262 If you look Gettinto, go later. Now, the second part of this question is asking us, Teoh find the Probably that the car has driven less than 800 months. It doesn&#x27;t anise and aid does. And so to do this, we&#x27;re gonna, um, find the Z value of Z score value for 8000 1000 minus, uh, 12 dozen for 94 over 1002 190 center deviation. We&#x27;re gonna get Dingell three point for eight. This is a corresponding Bilgi. Loves your 251 I believe. Choose your zero. That is her phone lunch. Because this is less than so. You don&#x27;t have to rearrange it. The first part of the problem Now the final part of this problem is asking, um, would you buy a vehicle if you&#x27;ve been told had been driven less than 6000 miles in the past year. And so we&#x27;re just gonna find the probability that car trip in less than 6000 Nos. And to do that, we&#x27;re gonna I need to do is find a Z value 6000 finest Wilson for for 90 90 for over was it doesn&#x27;t told her. 90 1002 119. That&#x27;s gonna give us negative sides. Sure, three for which has, of course, my brother gives you, which is very, very small. One very small.

Derrick Hanson · Answer

As you can see, I&#x27;ve got to normal distributions drawn out here. 1st 1 up top is generic. Meaning it&#x27;s just any time you were told that you have a data set that is normally distributed. This is the type of scenario that you&#x27;re looking at. Okay, We know from the empirical rule that 60% of our data is kept within one standard deviation. 95% of our data is kept within two standard deviations, and 99% of our data is kept within three standard deviations. I&#x27;ve gone ahead and split those out because we know normally distributed means that are mean or average is smack dab in the middle. So it&#x27;s splitting everything in half. You can see that 34 on 34 68. So on so forth. I am assuming that you guys understand a basic normal distribution before we&#x27;re going through with this problem. If all those percentages means nothing to you and you have not been taught H and Eric normal distribution, then you&#x27;re not going to be able to understand how to apply it to a specific situation like this problem. So before you continue with this video, make sure you understand how I&#x27;ve got those generic percentages before we start trying to apply it here? Okay, Now, assuming that you do understand those looking at this second normal distribution, this represents the actual problem we&#x27;re doing. We&#x27;ve been told we&#x27;re talking about the life of car batteries. We were told that the mean is 100,000 miles and that the standard deviation is 10,000. So I&#x27;ve gone ahead and marked out our three standard deviations. I added 10,003 times to the right, subtracted 10,003 times to the left to get my 1st 2nd and third standard deviation away from Army and of 100,000. Okay, Now the question here is asking us what is the probability that you will have a car that you can get a car battery that lasts between 80,000 and 100 and 10,000? Well, okay, let&#x27;s mark what they&#x27;re talking about. Then here&#x27;s 80,000. Here&#x27;s 100 and 10,000. So it&#x27;s asking us about this portion this interval of our chart of our data of our graph. Okay, if you understand the generic normal distribution and it&#x27;s percentages, this problem really isn&#x27;t that tough. All you have to do is add up each of those slivers each of those portions of the graph. So on the generic one, here&#x27;s the second standard deviation cause noticed. The 80,000 is on the red. It&#x27;s the second standard deviation. Here&#x27;s my positive first standard deviation because the 110,000 is the blue. It&#x27;s the first standard deviation. So we&#x27;re talking about this portion, meaning we simply just need to add all three of these percentages together. So that means we&#x27;re taking 13.5% from the red, plus 34% plus another 34% right. That&#x27;s the portion of the graph that we&#x27;re looking at here. If you take 13.5 plus 34 plus 34 that will give you 81.5% or remembering that percent means to divide by 100 81.5 divided by 100 meaning in decimal form, that would give us 0.815 That is actually all we have to do for this one, because it did just ask us for a probability we don&#x27;t care about the fact that the company makes 20,000 batteries a month because it didn&#x27;t ask how many batteries this time, it just asked, was the probability. So we&#x27;re done.

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Use the following data. The lifetimes of a certai…

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