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The National Health Statistics Reports described a study inwhich a sample of 550 newborn baby boys were weighed. Their meanweight was 3315 grams with standard devia...

Question

The National Health Statistics Reports described a study inwhich a sample of 550 newborn baby boys were weighed. Their meanweight was 3315 grams with standard deviation 567 grams. Anobstetrician claims that the mean weight of newborn baby boys isgreater than 3300 grams. Do the data provide convincing evidencethat the obstetrician’s claim is true? Use 𝛼 = 0.01.

The National Health Statistics Reports described a study in which a sample of 550 newborn baby boys were weighed. Their mean weight was 3315 grams with standard deviation 567 grams. An obstetrician claims that the mean weight of newborn baby boys is greater than 3300 grams. Do the data provide convincing evidence that the obstetrician’s claim is true? Use 𝛼 = 0.01.



Answers

The birth weights of full-term babies are normally distributed with mean $\mu=3400$ grams and $\sigma=505$ grams. Source: Based on data obtained from the National Vital Statistics Report, Vol. 48, No. 3
(a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of full-term babies who weigh more than 4410 grams. (c) Suppose the area under the normal curve to the right of $x=4410$ is $0.0228 .$ Provide two interpretations of this result.

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.

We're told in this question that babies that are born after a 32-35 weeks, just they, excuse me, gestation gestation period on average way 2600 g with a standard deviation of 670 g. And then babies who are born essentially full term at 40 weeks way 3500 g with a standard deviation of 475 g. So they weigh more but have less variation in their weights. We're told them that we have a baby who was born at 34 weeks, so falls into that first time period and wait 2400 g. And then we had a baby who was born at full term weighs 3300 g. So each of them way less than the respective average for their time period. But we want to know which baby weighs less in putting into perspective the different again, gestation periods that they had. So this is where we want to use the idea of Z scores. It allows us to take data that comes from that's represented by different parameters and compare them to see which baby does technically way less. Not just in terms of its weight, that's pretty obvious, but in terms of the average for that time period in which it was born. So first we'll look at calculating the Z score for the baby that was born at 34 weeks. Our formula for Z score is your observed value minus your mean divided by your standard deviation. And so for this particular problem, that's going to mean that we're taking the $2400. Excuse me, 2400 g. We're subtracting the mean that corresponds to 34 weeks, so that's 2600. And we're going to divide that by the standard deviation of 670. We're going to have a negative Z score. Which makes sense because our weight is less than the mean. So in our numerator we have negative 200 Divided by 670. To get a Z score of approximately negative 0.30 by round to the nearest 100. It's negative 0.30 So it's just 3/10. This baby was just three tents of a standard deviation below the mean. We want to compare that with our baby that was born at full term. So again, if I apply the Z score formula, I'm going to take 30 300 g And subtract 30 500. Again we're going to have negative 200 But this one is being divided by 475. So We can see they both weighed 200 g less than average for their gestation period. But we kind of want to know which one of them. Is that more unusual? Um and and kind of a greater less weight, if that hopefully makes sense? So now negative 200 divided by 475 gives us a Z score of negative 0.42 If you think of your normal curve, and we kind of take a look at each of these, I'm gonna look at them even though they wouldn't be on the same normal curve, because they have different means and standard deviations. If we just think of the idea that the mean would be in the middle, our baby that was born premature Would have been about here, let's say at negative 0.30 standard deviations away from the mean, and our Baby that was born full term would be there. And I'm gonna right above here at negative 0.42, it was slightly more to the left of the mean, meaning that that baby technically weighed less in respect to babies of its same gestation period at birth. So to answer the question, the Baby born at 40 weeks weighed less when compared to, oh sorry, again it weighed less when compared to um average babies at 40 weeks. Because it's Z score was more to the left.

If you happen to do problems seven, you're going to see that this one's very, very similar. There's just one slight difference that all be sure to point out. But we're talking about the weights of babies for those that are born slightly premature between 32 and 35 weeks of gestation period And babies that are born full term at 40 weeks. We're told that the mean wait for the Primi babies is 2600 g with a standard deviation of 670. And the mean weight of full term babies is 3500 g with a standard deviation of 475 g. We want to know which baby weighs less relative to the gestation period. If a baby that was born at 34 weeks, weighed 3000 g, so it's technically weighing 400 g more than average than the mean. And then kind of looking at the same thing for a baby that's born full term Again, g more than the mean, 30 900 g. But just because you can't answer the question just based off of that and say, well, neither of them way less. Um since they're both 400 g over the average weight, we also can't automatically say that the primi baby weighs less just because its weight was less. I mean, yes, it does. But it doesn't weigh less in relationship to the gestation period. We kind of want to know which one is more extreme from the average. Now, the one thing that makes us a little different from number seven is these weights are more than the average, Which would be a good thing, especially for the pre me baby. But we still want to know the idea of being less again when kind of compared to their average. So we're using the idea of Z scores so that we can compare them because they're kind of on two different scales. Even though we're talking grams, we've got one average of 2600 versus another average of 3500. So we can bring them to the same scale kind of re scale them. Standardize them by calculating the Z score. R Z score formula is your observed value minus the mean divided by the standard deviation. So our Z score for the Baby that was born at 34 weeks will be 3000 minus 2600. Over 670. was the weight of the baby at that time period. The mean is 2600 with a standard deviation of 6 70. So our numerator is going to be 400 if I subtract those And divide that by 6 70. Rounding to the nearest 100th because these scores are usually carried out to two decimal places. I get .60 and then I'll do the same thing, calculate the Z score for the full term baby That had a birth weight of 3900 Compared to its average of 3500, Divided by the standard deviation of 4 75. So again we have a difference of 400, but now we're dividing it just by 475. So we get a Z score of .84, so which one weighed less? It would be the one in this case and this again is going to be a little bit different in the number seven. We had negative Z scores, so we kind of look at those a little differently, but this one is going to be in terms of how close it is to our mean, if we look at our normal curve, even though these would both be on a different normal curves, since they have different means are mean, is always in the middle of the normal curve. Our premium baby was six. Standard deviations above the mean. That's what the Z score is. It's a measurement of how many standard deviations away from the mean. Is that observed value? So, this one would be .6 standard deviations above the mean, Where the baby that was full term was slightly higher than that at .84. So it's always when we're talking about which one is less, it's always the one to the left, which means in this case the early term baby weighed less Or the 34 week. Yeah, weighed less relative to its gestation period. Mhm. Again, because the Z score was further to the left, we could look at the area. If you were using a Z table, you could look at the area that's to the left of that, And that would give you a smaller area than the .84. The full term baby

Let us look at this question. Now we're here. It says that we have 1 95 boys. The sample mean and the sample standard deviation is given. And we want to use a 95% confidence level. And it says are the results very different from those found in exercise? Nine. Well, in exercise nine, these were the results. And over here, now we're getting the values. If we put it in the software, if you will get 31.76 to 33.63 31.76 2 33 something so to 33.63 So, yes, they are very different. And does it appear that boys and girls have different ways? Yes, definitely. We can see that these weights are a little higher than as compared to the girls. So, yes, they are different and they do have different rates.


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