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Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from two to six pounds with a mean of four pounds and standard deviation of 1.1547. We randomly survey 64 homes with children.
a. In words, $X=$ ______
b. The distribution is ____.
c. In words, $\Sigma X=$ ______
d. $\Sigma X \sim$ ____(____,____)
e. Find the probability that the total weight of open boxes is less than 250 pounds.
f. Find the 35th percentile for the total weight of open boxes of cereal.

All right, now this is problem number 33. Now, the birth weight of full term babies are normally distributed with a mean new as 3400 g. Right? So they are normally distributed lettuces to draw a normal distribution. You can use a statistical software for this, but I'm just using doing this by hand. Right. So this is Mu is equal to 3400 g, right? And the standard deviation is 505 g are sigma is 505 g. Right? What will be the point of inflection? Points of inflection are where the curve activity in this excavator. So the point of inflection Arm U minus Sigma and mu plus sigma. We have both of these values. So what would you mean? Minus sigma 3400. Minus 505. So it says +2895 So this is to 895 and new plus 5 five is going to be what? This is +39 05 Right. These are the points of inflection. Now, the first one is draw the normal curve with the parameters. Yes, we have already done that. Then shared the reason that represents the proportion of full term babies who were more than 4410 We're more than 4410 which means 4410 is actually going to lie to the right of this, right? 4410 If I want to label 4410 happens to be two standard deviations of the rain because I add 505 in this once more, this is going to give me 44104410 And this is the area that we are interested in. That is, to the right of 4410 right shade the reason that represent the proportion of full term babies who were more than 4410 g. All right, so part B is also done. It is two standard deviations away to the right. But since I suppose that the area under the normal curve to the right of 4410 is 0.28 So let's say that this shaded region is 0.2 to 8. What does this mean? This means that 2.28 percent of the babies these are the birth weights of full term babies. What? What exactly is full term babies? I don't know, but okay. This means that 2.28% of full term babies full term babies, the more than 4410 g. This is one interpretation or another interpretation is 97 point 72 percent of full term babies. How full term babies? They less then 4410 g. Right. Why 97.72? Because if this area is 2.8% the rest of the area that is to the left is going to be 97.70%. So these will be the two interpretations.

Welcome to new Madrid. In the current problem we are considering the weeks of babies. So here wait off newly born piece. Okay. And also these weeds are distributed normally with mean it follows normal with mean, mu is equal to zero g and sigma is equal to frank 05 lamps. Okay, so first we want to see the normal distribution for this. Now if we know the normal skill goes like this New equals 23400 New minus sigma. That is 3400 minus 505 And mu plus sigma. That will be +3400 plus +50 five. New plus two sigma will be either this value plus another sigma. What simply substitution? Two times 505. Okay, so this is the way we construct now. If we just bring the normal distribution. Okay, It is going to be like this. And then we can put 3400 over here. And if we add 505 we get 3905. Okay, if I had 505 again with this we get zero one, find 44 Correent? For 410. And then again, if I had 505 I get five move on 94 correct. So I will remove the extra wordings over here headache. Now there is another way also we can use that is yeah, Equals 23,400 -505 because but I think is this but it's a disk over here. We will have to eat everything. I mean if i subtract 505 here we get zero nine three two got it. So if I do here two things this To get training. If I do three times We had 1 18 claim that yes, This one is 1885. All right, so we can go ahead and clear this. No. We are asked the next question to be. So this is the distribution. So you can say this to be the mean. Oh sorry, is to be the mean and this dis tends to be sigma distance. So this is a mule plus sigma nu minus new plus two sigma and so on. No. The second question that is asked to us is to share the region for her to saw the baby is the proportion of babies who were more 4410. Great. Okay, so you need to find this beauty. So if you see this is where for 410 East. Well this regional well here these yes, that is required. So it is something like this shaded vision right now. The third question we're here is supposedly under the normal curve to the right of excess equals to 4410. It is this area is given to 0.1587. Okay, so this is going to be 0.1587 provide to interpretation. Now, if you see 1587 would mean almost Okay, let's let's make it 16% or you can leave it. Uh 15.87%. Also, whatever is fine for you. I realize that now for explaining purpose, I will use 16 because that's faster. It's easier to see. So the first interpretation that we will have here is since only 16% of all babies lie above 4410 g. We and conclude that this is not a very likely event, But it only 15%. So it's a rare event. Okay. On the other hand, we also find that 100 -16. So that would be something around 84%, That is. Majority of the babies born will have whips Less than 4410 grams. That means it is It is rather than Generalized value. And yes, definitely some Children will have with more than 4410. But that's not a lot. If that is happening, if that is happening a lot, then either the distribution does not follow normal or there has to be some situation because of it. The values are happening to be concentrated there. So I hope you could understand good explanations. One is we explained what is the meaning of 16 for us and the other is what is the meaning of this remaining 84% for us. So I hope you could understand this. Let me know if you have any questions.

In this question, we are given the following probability distribution that I have run out. And Paul A Also, this Is this a discrete probability distribution on the answer is no. Because of the five parts. So here we have five plus and that five plus could be anything from 5.5 to hundreds to thousands. Anything. Obviously, thousands of Jordan isn't that realistic. But anything about five is counting that, which is why it's not discreet, and that's dies the altar film to be else just to draw a history Graham for this data. So gonna draw that hair cause charges dropped just around me. So we've got one 01 to three full on five plus who go there. Want to three? No, but plus so do a history room with an interpreter in up her sense, which are probabilities. Times 100 on we can see the biggest one is 24. So do five, 10 15 2020. But you don't always need to go away to 100. Um, because in this case, that would be better waste full of space because nothing goes above 25 so zero is no point. Turn I'm, which is just over 20%. So about that. And when you do yours probably don't on graph paper with a ruler. So it's the lining Will actor in the month? So it was my mistake. Here she goes upto 38 se 25 30 that fuck. So the next one is 38 which, if that's 30 that's that Fivethirtyeight, the Murray bugger off the next one is 24.9, which will be about just on the 25. Three is well, 10.6 just above 10. Four is no point north three to So that's only just 3.2% and then a five close is even smaller. So that is a shame for a graft. If we did a little shape, we get something that looks a little bit like that if you else to describe the shape you consent, has Askew Teoh towards the room one and stuff it up. So that is pub be so now we're placing five with exactly five. That's five plus, so we're going to scrub that there and just say this is five. Otherwise, this next It's impossible. So we think asked to find the mean under standard deviation. So if we rewrite all table down here with zero one to three full, I remember this time when just putting five. So if we put on our poll Liberty so no point to a nine no 0.384 No 0.2 for nine. No 0.1 I six. No point North 32 on. No point, no to. So the first thing we need to do is do X comes PX to get the mean. Does he ever times anything is zero on one times. Anything remains the same. So two times no 20.2 for tonight gives us no point for nine eight and then three times. No 30.106 gives us no 0.318 four times not point North 32 because there's no point. Want to eight on five times No point Not too. Oh gives us no 0.1 to find the main. We need to sum everything in that road. So we've got zero very plus no point 38 full, plus no point. For +98 It's no 0.318 plus no 0.1 to 8, plus no 80.1 which gives us one. Well, that should be a decimal point, that one point for two eight. So that is ah mean, which is also known as the expectation of X. The next thing we have to do to get our standard deviation is find the variants. For that, we're gonna need X squared P X, the first to remain the same. Cause 00 and one squared is one. So two squared is 43 times that by not 430.2 for nine, we get no 0.996 three squared is nine. And if you times that by no 90.106 we at no point no for four Foursquare's 16 on that times no point North 32 gives us no point 512 five Gold is 25 if you times that by no point not too Oh, we get no 0.5. So now we want to some all of those ups we've got our no 0.384 There's no 0.9 at six. That's no 60.954 That's no 0.512 plus that last 4.5, which gives us 3.34 eight. So we first warning are variants, which is a of X squared, which is the value we found. Yeah, so we can write that below 3.348 minor e of X Square, which is on mean so 1.4 to 8 squared so that 3.3 for eight minus 1.4 to 8 squared, which is two point 039 So then if we do that some of the 3.348 to minus 2.39 gives us 1.39 and the last step this two square root bum, which gives us 1.1 for cool and that

1 36 So for part A, you want us to verify so the probability of 0 17 80,000 447 27 Divided by the total 9 to 936 Probability. 18 24 A 855 0.7 divided by 29 to 9 36 25 2 34 3938 to 5 Very by very nice 66 six to ground three equals 1 to 3% equals 29%. All right, that's a verified part. So Part B 18 to 24 It's the same question. The credibility. 18 24 instead, percent R C. They want older than 17 older than 17. I'm going to do the, uh, one minus 0.25 0.75 which is 75%. So if you take up and you add the 10% 13% 23% and the 29% you'll get the same 75% her d. We're looking at between 18 to 24 or older than 70 so between eight no and between 18 to 24 older than 17. So both of those criteria have to be true. So the only thing that makes them both through is between 18 and 24. And that would be 10%. He is my or so it could be they could be 50. Plus, that would satisfy the or so anything in that, um, any category bad. The 0 to 17 makes that true. So that would be my 75% one minus the 25%. Uh, I added for F I added the 13% lost 23% plus the 29% and I got 65% and G is no more than 24 so that has to be 24 or less. So that would be 25% plus 10% equals 35 person.


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