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Question 5Go5etsThc Illcepan ol rcfrigetators ratmally distributed Fitnn 14ycars and standard devation 2,7 ycars If14 refrzerators randomly selected, there %a 724 p...

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Question 5Go5etsThc Illcepan ol rcfrigetators ratmally distributed Fitnn 14ycars and standard devation 2,7 ycars If14 refrzerators randomly selected, there %a 724 probabillty that the mean lifespan of the selected refrigerators betrieen what two values? there ale Lwo answers, IIst both answcts scparatcd by conmn

Question 5 Go5ets Thc Illcepan ol rcfrigetators ratmally distributed Fitnn 14ycars and standard devation 2,7 ycars If14 refrzerators randomly selected, there %a 724 probabillty that the mean lifespan of the selected refrigerators betrieen what two values? there ale Lwo answers, IIst both answcts scparatcd by conmn



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The lives of refrigerators are normally distributed with mean $\mu=14$ years and standard deviation $\sigma=2.5$ years. Source: Based on information from Consumer Reports (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of refrigerators that last for more than 17 years. (c) Suppose the area under the normal curve to the right of $x=17$ is $0.1151 .$ Provide two interpretations of this result.

Right. The average time spent sleeping in ours for a group of medical residents as a hospital can be approximated by a normal distribution. So we're going to draw our normal curve And you had to look at the figure to find that the average or the mean was 6.1 and the standard deviation Was 1.0. So for part a we're trying to find the shortest time spent sleeping that would still place the resident in the top five of sleeping times, so the top five would be right here. So now, if there's 5 to the right of that boundary line, that means there's 95 to the left. So ultimately we're trying to find the sleep time in hours. But in order to get there, we are going to have to find the Z score first. And the Z score that is associated with. That can be found by using your inverse norm function on your graphing calculator. And when you use inverse norm, you have to provide the area in the left tail, followed by the mean, followed by the standard deviation. So in our left tail we've got .95, Not a .05. Since we are finding a Z score, the mean of a Z score, or the standard normal curve is zero and the standard deviation is one. So I'm gonna bring in my graphing calculator and I'm gonna do second. There's to access my distributions functions and it's number three in my menu. So the area in the left tail followed by the mean of the standard normal curve. And the standard deviation of the standard normal curve gets me a Z score of approximately 1.645 So that means on the Z scale now remember the mean is zero, So we just found a Z score. So right here would have a Z score of 1.645 So our goal is to find that X value. So we need to transition back. And the way we'll transition back is we'll use the formula you have for Z scores. And you know, to find a Z score, you take x minus mu divided by sigma. So if I were to use some algebra skills and calculated my cross products, I would end up with X minus mu is equivalent to Z times sigma. If I then added mu to both sides and isolate the X. I've got a new formula that says X is equal to mu plus Z times sigma. So I'm going to apply that formula to find the boundary line That separates the top five of sleep types. So our average was 6.1, We found the Z score to be 1645 And our standard deviation was 1.0. So that ends up where 7 745 hours Would be the shortest time spent sleeping. That still places a resident in the top five of sleep times. So that was part A. Now let's go on to part B. So in part B we're going to draw the same curve, We're still going to have a mean of 6.1, but this time it wants to know About the Middle 50%. So we're going to draw a line here and a line here And say that this is 50 of the curve. So now the whole entire curve would be 100%. So if 50 is in the center, that means there's 50 left over to be split evenly between those two tails, which would make the left tail to be .25 And the right details to be .25. Again, we're going to have to find the Z scores affiliated with each of those boundaries. So to find this Z score, we're going to do in verse norm. Again, it's the value in the left tail, so left of this line Would be .25. The mean of the standard normal is zero and the standard deviation is one. So we bring in my calculator, second vares Inverse norm of 2501. And I'm going to get a Z score of approximately negative .67. So this is the score right here is negative 0.67 So now I want to find the Z score on the other, the right hand boundary. So I'm gonna use Z equals inverse norm. But this time from this line into the left tail would be 75 of the curve. And again our standard deviation and our average didn't change. So I'm going to bring in the calculator, inverse norm 0.75 With a mean of zero and a standard deviation of one gets me of Z score of positive 67 Yeah. So now we want to find each corresponding X value. So the first one and do this one, we're gonna say X equals mu plus Z time sigma. So arm you was 6.1. The Z score associated with that boundary line was negative 0.67 And the standard deviation 1.0 Gets me seven No, sir, it gets me five 43 hours. And then doing the other ones were going to do this one right here, X equals mu plus Z times sigma. So we're going to get 6.1 plus the Z score on this boundary line was positive .667. And we will get 6.77 hours. So just to recap part a what is the shortest time spent sleeping? That would place a resident in the top 5%. That's going to be 774, 5 hours and part B. Between what two values does the Middle 50 of the sleep times lie, The middle 50% is going to lie between 543 hours And 677 hours.

Welcome to New married. In the current problem we are given that life of repeated refrigerators is the variable under consideration. So life of I will say fridges. It's easy and short than refrigerators. And this follows a normal distribution with new is equals two 14 years and cigna is equal in school people. Right? Yes. No. First till you got here in the last days to draw the cough. Okay. And second we have to find that the regional that is less than equal to 17 years. and the 3rd 1 is that Suppose this probability of excellence and equals 2 17 is equals two 0.1151. Then what does it mean if this. Okay bagram first itself we will get this. We're here. Okay until you do if you see The distribution has centered at 14. So this will be about 14. So So this is 14 and then 14 -2.5 with this because 14 -2.5 that will be one sick Magistrates, correct? Each of these are sigma sigma sigma. Today is minus three sigma and it is lustrous sigma plate. So this would give us 11.5. Then again if i subtract 2.5 I get nine Again if i subtract 2.5 I get 7.5. Now for 14 if I add 2.5 I get 16.5 Again if I had 2.5 I will get Yeah. Uh huh 19 If I again add I will get 21.5. Now If you see 1717 will lie somewhere over here. Got it. So we had 17 over here. So therefore in this peace if you see what is the probability at it is more than 17 years. So of course this should not be the sign. It should be greater than equal to that is the right side of the distribution. So this is the media that is she did. Okay. This is part of are A. Is this diagram. And of course in the diagram you don't have to show this. Yes, everything you should show. So this is a diagram. B. And now part B. Part beast. Part C. Is telling. Okay, if this this probability is 0.1151, What do you interpret out of it? So for interpretation, what we can do is we can utilize this case to write the interpretation. So the first interpretation is refrigerators exceeding 17 years of life is uh quite rare phenomena, right? Because as out of 100% only 102 0.1151. The physicals to 11.51 experiences product only out of only experiences that long. Yeah, lifetime. Okay, So it can be thought like somebody using it with too much of care and responsibility. Only those people will be able to have or maybe the machine will be so good that it will be going beyond 17 years. And what is other interpretation? The second interpretation? The second interpretation is the fridge is the refrigerators are expected to the fridge? Is us expected to function still 17 years As 100 -11.51 percent. So how much will that be? Nine four eight. Eat. Okay, so 88%, That is near about 90% writes. The majority of the population is below This. So they are expected to have a like a maximum teen 17 years. So from this one number we can look at the entire dark uh data set from two point of view, one is greater than one is the less than okay. What if the less than values what if the greater than values. So I hope you're able to understand this. Let me know if you have any questions

In this problem, we're told that the life of a refrigerator is normally distributed with a mean of 4.8 years and the standard deviation of 1.3 years. And in part they were asked if the machine is guaranteed for two years. What is the probability that you would have to replace it under warranty if you bought it so it's replaced under warranty? If it lasts for less than two years? We're looking for the probability that X life of the machine is less than two and using the said conversion if that is Equity X minus mu over Sigma. So this is equal to the probability that said is less than to minus 4.8 over 1.3, and that comes out to 0.156 So the probability of replacing it under warranty if you buy it is 0.156 on a part B were asked what time the manufacturer should give as the guarantee if they only want replace 0.5%. So let's say the guarantee is que years, so if if the time that the machine last is lesson K years they replaced it under guarantee, and they want the probability that occurring to be half a percent. So that's the probability that the life is less than K years is equal to zero point 005 That's half a percent. So we can also view this as the probability that Zed is less than K minus 4.8. Over 1.3 equals zero point 005 And using either the standard normal tables or a calculator or computer software, you can find that the correspondent corresponding Zed score is negative two point 576 which implies that K is equal to one point 451 years. So you want to set the warranty at 1.451 years if you only want to replace half a percent of them.

At the time it takes to wait for a kidney transplant For people in the age group of 35 to 49 is approximated by a normal distribution. So we're going to draw that normal curve. And when you looked at the figure, you found that the average wait time was 1,674 days. With a standard deviation of 212.5 days. So part a is asking you to find the wait time that represents the 80th%ile. So when you're talking, in terms of percentiles, if you score at the 80th%,ile, that means You did better than 80 of the people that took the test. So that means 80 scored lower. So when we're talking the 80th%,ile, we're talking about 80 have a lower weight time in terms of the kidney weight. And if there's 80 on the left side of this boundary line, that means there's 20 that are on the upper side. So the first thing we want to do to figure out the boundary here between the lower 80 in the upper 20 is we're going to find a Z score and the fastest way is to find that Z score will be inverse norm in a graphing calculator. And when you use that function from the calculator, you do need to provide the area in the left tail, followed by the mean of the standard normal curve and the standard deviation. So for our problem, the area in that left tail is 80 of the curve. The mean for a Z score, the Z scores, the standard normal scores and the mean of Z scores is always zero and the standard deviation is always one. So I'm gonna bring in my graphing calculator and I'm going to access the inverse norm by hitting two and the vares button, which gets me into the distributions menu, and it's number three in this menu. So the area in the left tail, followed by the means of z scores, followed by the standard deviation. We end up with a Z score of approximately .84. So we're saying that this boundary is a Z score of about .84 And keep in mind the mean is a Z score of zero. So we now need to find, how many days does that represent. So you have a formula that says Z equals x minus mu divided by sigma. So we're going to take that formula, we're going to use our algebra skills and we're going to transform it so I can perform cross products here and get x minus mu equals z times sigma. I can then add mu to both sides and say that X is the same as new, plus Z times sigma. So I want to apply that formula. So I'm going to say mu plus Z times sigma. And the average wait time to get a kidney was 1,674. The Z score I just found was .84 And the standard deviation provided was 212.5. And when I calculate that out I will get 1 852 0.5 days, Part B in part B. We want to know what waiting time represents. The first quartile. So again, I'm going to draw that bell shaped curve And in terms of court, als, The first quartile represents the 25th%ile. So again, that's saying 25 is lower than that boundary line. We still have our average at 1,674 days. And again we are going to find the Z score attributed with this boundary line by using our inverse norm. The area in our left tail is .25. The mean of the Z scores or standard normal curve is always zero and the standard deviations one. So we'll do in verse norm 0.25 comma zero comma one. And we're going to get a Z score of about negative 10.67 So we want to find the raw score for that boundary line. So again, we're going to use the formula we derived. So we're going to take our average which is 1,674 days plus the Z score associated with this boundary. And multiply that by the standard deviation of 212 5 days and we will get 1531.625 days. So let's just recap What waiting time represents the 80th%ile. So that's saying about 80 of the time you're waiting, or you're you're waiting less than 1,852 5 days and 25 of the time You're waiting less than 1,531 625 days.


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