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.