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If the duration of a human pregnancy is approximately normalwith a mean of 266 days and a standard deviation of 16 days.(a) What is the probability of randomly sele...

Question

If the duration of a human pregnancy is approximately normalwith a mean of 266 days and a standard deviation of 16 days.(a) What is the probability of randomly selecting a woman andher pregnancy lasts less than 277 days? Round to the nearestday.(b) At least how many days should the longest 25% of allpregnancies last?(c) Suppose a certain obstetrician is currently providing careto 10 pregnant women. What is the probability that these women willhave a combined pregnancy length greater than 2,730 d

If the duration of a human pregnancy is approximately normal with a mean of 266 days and a standard deviation of 16 days. (a) What is the probability of randomly selecting a woman and her pregnancy lasts less than 277 days? Round to the nearest day. (b) At least how many days should the longest 25% of all pregnancies last? (c) Suppose a certain obstetrician is currently providing care to 10 pregnant women. What is the probability that these women will have a combined pregnancy length greater than 2,730 days?



Answers

The length of human pregnancies are approximately normally distributed with mean $\mu=266$ days and standard deviation $\sigma=16$ days. (a) What percent of pregnancies lasts more than 270 days? (b) What percent of pregnancies lasts less than 250 days? (c) What percent of pregnancies lasts between 240 and 280 days? (d) What is the probability that a randomly selected pregnancy lasts more than 280 days? (e) What is the probability that a randomly selected pregnancy lasts no more than 245 days? (f) A "very preterm" baby is one whose gestation period is less than 224 days. What proportion of births is "very preterm"?

Okay. This question says that the lens of human pregnancies are approximately normally distributed with mean of 2 66 days and standard deviation of 16 days. So let us draw the normal curve. The mean is to 66 the standard deviation is 16 days. Now. What proportion of pregnancies last more than 2. 70 days to 70 will fall somewhere around here. This is going to be to 70. So what I want to do is I will find the corresponding Z statistic and find the P value, which means the area and the still it ain't. And what is the question saying? The question is saying they last What proportion last more than 2. 30 days. So this is exactly what I want. So I want to find a P value, right? So what I will do is I find the statistic for to 70. And what is the formula for that statistic? X X minus mu Bye, sigma. So part A. We'll find the Z statistic for to 70. This is going to be 2 70 minus 2, 66 upon 16. So this is nothing but four by 16 or this is going to be 160.25 So what is the fever? You can either use a set table for this or the statistical package like SPS or are or accept. The thing is that if you use a certificate package, it will give you an exact answer. So I'm using an online tool here, and my P value is 0.401 So what is the probability that X is greater than 2. 70? It is 0.401 Right, Moving on the part B. What proportion of pregnancies last less than 2 50 days. Okay, so less than 2. 50 means to 55 somewhere around here. So less than means I find the statistic for 2. 50 and find the P value for them, which is going to give me the area and the tail. So let us do that. Mhm. So what we want is zero for 2 15, right? Does that value for 2 50? So this is going to be 2 15 minus 2, 66 upon 16, which is going to give me minus one. So what is the P value? Oh, minus one. I just put in minus one. And this p value is 10.1586 Okay, so this is this corresponding value is 0.1586 So what is the probability that my ex is less than 2 50? This is going to be point 1586 We're moving on to part C. But she says what proportion of pregnancies last between 2. 40 and 2. 80 days, so to 40 and to 80 days 2. 40 somewhere around here to 80 is somewhere around here. So what I will do is I'll find the corresponding Z statistics for these two. I will find the P values, which means for 2. 40 the area will be in this tale. And for 2 80 the area will be in this tale. I want the central area, which means one minus both of these values. It will give me this central area. This is what I want, right? So let us try to find this. Okay, so we once said for two. 41st, this is going to be 2 40 minus 2, 66 upon 16, right? Yeah, I'm 1 16. This is going to be 26 by 16. So this is 26 divided by 16, which is 1.625 1.625 Now, what is the corresponding P value for this? The area and the left him. I use the P value calculators and this is one 0.625 had entered. And that is 0.5 This is 0.52 Similarly, what is it That statistic for 2. 80 it's 2 80 raid that we want. Yeah, For 2. 80 this is going to be 2 80 minus 2, 66 by 16. So this is going to be 14 by 16. So 14 divided by 16. 8.8750 point 875 What is the corresponding p value for this? 2.875 0.875 I hit Enter my p values 0.1907 So this is 0.1907 So I add these twins of practice from 11 minus 0.5 two last 0.19 zero seven. Right, So this is nothing but one minus zero point 24 27 If I am not wrong, right, this is going to be seven to this is going to be four and one. Yeah. So this is going to be 75 0.734 point 7573 rather 0.7573 This is 0.7573 This is the probability that my ex is between 2. 40 and 2. 80 now moving on to part D. What a spot. But do you have to say probability that the randomly city pregnancy lasts more than 2 80 days now more than 2. 80. So what is the P value that we found for 2. 80? It was 19.7 So, probability that any pregnancy lasts more than 2 80 days is going to be 800.19 07 Moving on to Part E. What is the probability that randomly selected pregnancy lasts no more than to 45 days, no more than 2 45 days, Which means I want probability that my access less than equal to 2 45. Okay, so we find the corresponding that statistic for 2 45. So this is 2 45 minus 2 66 divided by what was Sigma 16. Okay, 16. It is minus one point 3125 So there's that statistic is minus 1.312 five. What is the corresponding P value for this? Minus 1.3 Went to five, minus 1.3125 ahead. Enter and I get 0.94 Sorry. 0.9476 point 09476 So I would say that this is equal 2.9476 I'm sorry about that. This is the probability that we want now. A very preterm baby is the one whose gestation period is less than 2 24 days. Our very freedom babies unusual or 2 24. Okay, if I say that this is 2 66. This is the mean. This is going to be 2 51 11 standard deviation away. This is going to be 2 34 the institutional deviation away. And this is 2 24 which means it lies somewhere around here. 2 24. Yes. So this we can say this area beyond 2. 30 is only 0.25%. And this is even 10 lower than that. So yes, this, we can say, is, uh, very unusual observation. So, yes, it is unusual. Yeah. Mm.

In Probleble 54. The lens of preferences from conception to birth paris according to an approximately normal distribution. This normal distribution has a mean of 266 days. And the standard division of 16 days. We have said that it's approximately normal distribution for birthday. We don't use a graphing utility to grab the distribution, distribution is normal. Then it has a function. One divided by sigma, blood by school route to buy deployed by eaters about minus x minus mu or square, divided by two sigma squid. We just enter sigma and mule in the graphing utility. Because I've entered dysfunction before. That's refusing christmas. You put sigma equals 16 and new equals 206. 6 in the function here. Then when you grab the function, we have that here and this is the function. And we can increase the scale to make it -1 Point Home. Two point. This is the distribution for barbie. We want to use an integration to approximate the probability that The pregnancy were lost from 240 days to 280 days. Then we want to get the probability for ex where X. Is the broken insolence to be between 240 And 208. This This integration is from 240. for F of X. The X. F. Of X is defined in this most. Then we need to enter the limits for this. Integration from equals 240 to be equals 200. And it's we can see that Approximately equals 4.75 steps. He says they Equals 0.757. We both see we will use this graphing utility or the integration utility To get the probability that Nancy will lost more than 200 and 80 days X. to be more than 280 days equals the integration From 280 to Infinity. For f of X. Dx, there's a change of integration limits to make it From 280 two infinity infinity. We can just put very large number. We can see that It goes from 280 to infinity equals a point 191 4.19 What? And this is a final answer. Marcy this for barbie and for body it's a graph.

So what do we have here? Well, the length of human pregnancies from conception to birth varies according to a normal distribution. Where mean is 266 days on the standard deviation is 16 days, which means sigma is equal to 16 days. All right now, we haven't given a few scenarios. Let us look at part eight part aces at what percentile is a pregnancy that last 240 days. Which means that X in this case is equal toe. 240. What are we going to use? We are going to use a zero distribution over here right as their statistic. How do we get a sad statistic? Z is equal toe X minus mu by sigma. So in part, a R X is equal to 40. Arm you is to 66 Sigma is 16. All right, So what is going to be my said in this case? It is going to be to 14 minus 2, 66 upon 16. If I use a calculator 2 40 minus 2 66. Just a moment to 14, minus 2 66 is minus 26. Divided by 16 happened This directed by 16, happens to B minus 1.6 to fight. So my Zen statistic is equal to minus 1.62 Faith. What exactly does this mean? What are they asking me? They are saying at what percentile is a pregnancy that last 240 days. So I want to find the person type. If I look at this distribution, this is a normal distribution, right? This is zero Verma's. That statistic is equal to zero, which means that this is at 58% time. What is my Z value In this case? My Z value is minus 1.625 which means that it falls somewhere around here minus 1.6 to 5. So in order to find what percentile this is on, I can simply find the P Valley, which means I will find the area in the state. Right. I can simply do that. How do I do that? I can either use a Z table or a statistical software. Anything will work. So I'm going to use an online tool. This is a set table. But over here we also have a very easy visualization. What was as a sadistic minus 1.6 to 5. So if I go over here around minus one point 62 Okay, this is just around six to we don't have 6 to 5 in this case. Okay, so this is yeah, minus 1.62 But we want minus 1.625 We can either use this or we can go down over here. 1.62 I'm looking at the rope off 1.6 and column off 0.2 So this is 0.4474 So let me just tell you what this 0.4474 is this 0.4474 is actually this area 0.44 point 4474 is the area between zero and this point, but we want the area to the left of this. We will surprise this from 0.5, right? If I do this, if I do a point by minus 0.4474 I get 0.5 to 6 0.0 five 26 So these many percentage off data points are below it. Sorry. These This is the proportion of data point that is below it. Which means that 5.26% data points are below it. So what is the percentile? 5.26 is going to be our answer if we just check this so we can write this as fifth percentile. Okay, this is nothing but the fifth percentile. The fifth percent time. This is how we actually write this. Moving on the part B. What does part we have to say for us? I think part visas. What percent off? Pregnancies last between 2. 40 and 2. 70 days. Okay, so now between 2 40 do to 70 days. Okay, If I standardized the values, this is my novel distribution. We already have the answer for 2. 40. Right? What is the C statistic for? 2 40 minus 1.6 to 5. So minus 1.65 with somewhere around here minus 1.625 and 2 70 is just more is just a little bit more than a mean off to 66. Which means it will fall somewhere around here. But what is this value? Let's try to calculate this. Value are the statistic is going to be to 70 minus 2 66 Upon our sigma, What was Sigma Sigma was 16 upon 16. So this is four by 16 or one by four, one by four is what point to fight 0.25 So this value this over here is zero. And this is 00.25 again. Let us take the help off this table. Now 0.25 means I'm looking at the row off 0.2 and the column of 0.5 So my answer is zero point 0987 Now, what is the 010987 It is the area over here. This is 0.987 Right, So 0.987 plus I want this area. Plus I want this area. And what is this area? This area I had already calculated over here as 0.4474 So if I just are these two less 0.4474 If I just add these to let me use a calculator for this 0.4474 plus 0.987 This is 0.5461 point 5461 or I can say 54.6 1% off. Data points actually fall in this region. Eso if I wanna frame this correctly 54.61% off Pregnancies actually last between 2. 40 and 2 70 days. So these many percent off pregnancies off pregnancies last between? Yeah, 2 42 70 days. All right, What is Park Si, Part C says how How long do the longest 20% off the pregnancies. Last longest, 20%. Meaning, What is this question trying to say? Let us this Look at this. If this is my normal distribution, they want this area. They want this area to be, uh, to be point to They want this area to be pointed. So this is 0.2. So the question is, how long do the longest 20% off pregnancies last? Which means I want to find what is going to be my Zen statistic over here. So I have my p value. If I say that this is zero. What should this area? This area should be 0.3, right? This should be 0.3. So if I just look at this table. Where can I find 0.3? If I look in the rose 0.8, I am getting close to 0.3 at zero point 84 So this should be around 0.840 point 84 Okay, so now I have my eyes. A statistic. I remember the formula. The formula was that is equal toe X minus mu by sigma. Now what all do I have? I have my zed value, which is 0.8 for 0.84 I have my immune, but just to 66 and sigma also it's a 16. So this is to 66 and this is 16. I have all of these values and accepts what I wanna find. Right? So if I use my calculator for this, this is going to be 0.84 multiplied by 16 which is 13.4 plus 2 66 which is to 79 to 79 0.44 This is my ex right? These many days. Okay, so the longest 20% of the pregnancies are going to be These many days are beyond this, right? Like like to 80 25 to 90 days. So this is going to be my answer for part C if I just check the answer. Yes, it is going to be more than 2 79.44 days, and this is how we go about doing this question.

Hey explores the one you might hear, so we're given the mean that is equal to 266 and the standard deviation, which is equal to 16. So we're just gonna plug this into the density function for a normal distribution of a random variable X with me and standard deviation. To get F of X is equal to one over 16 square root of two pi e to the negative X minus, minus 266 square over 512. So the probability of a female to have a pregnancy that last 252 days to 280 days, which is 36 40 weeks, respectively, is gonna be the probability between those two numbers 252 and 280 which is equal to then to go from 252 to 80 and we just plug in. This equation off off of X is equal to one over 16 square root of two pi e to the negative X minus 2 66 square over 512 T x, which is around 0.6 money. So we expect around 618 females


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