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16.26Bone loss by nursing mothers. Breast-feeding mothers secrete calcium into their milk Some of the calcium may come from their bones_ so mothers may lose bone mi...

Question

16.26Bone loss by nursing mothers. Breast-feeding mothers secrete calcium into their milk Some of the calcium may come from their bones_ so mothers may lose bone mineral Researchers measured the percent change in mineral content of the spines of 47 mothers during three months of breast- -feeding: Here are the data: BONELS-4.7 -2.5 -4.9-2.7 -0.8 -53 -8.3 -2.1 -6.8 -4.3 22 -7.8 -3.1 -L.0 -6.5 -1.8 -5.2 -5.7 -7.0 -22 6.5 -L.0 -3.0 -3.6 -5.2 -2.0 -2.1 -5.6 -4.4 -3.3711~4.0 -4.9 -4.7 -3.8 -5.9 -2.5

16.26 Bone loss by nursing mothers. Breast-feeding mothers secrete calcium into their milk Some of the calcium may come from their bones_ so mothers may lose bone mineral Researchers measured the percent change in mineral content of the spines of 47 mothers during three months of breast- -feeding: Here are the data: BONELS -4.7 -2.5 -4.9-2.7 -0.8 -53 -8.3 -2.1 -6.8 -4.3 22 -7.8 -3.1 -L.0 -6.5 -1.8 -5.2 -5.7 -7.0 -22 6.5 -L.0 -3.0 -3.6 -5.2 -2.0 -2.1 -5.6 -4.4 -3.3 711 ~4.0 -4.9 -4.7 -3.8 -5.9 -2.5 -0.3 -6.2 -6.8 1.7 0.3 -23 0.4 -5.3 0.2 -2.2 -5.1 The researchers are willing to consider these 47 women to be an SRS from the population of all nursing mothers. Suppose that the percent change in this population has standard deviation 0 2.5%. Make a stemplot of the data to verify that the data follow Normal distribution quite closely. (Don 't forger ~ that you need both a 0 and a -0 stem because there are both positive and negative values:) Blend Images/Superstock Use 99% confidence interval to estimate the mean percent change in the population.



Answers

Bone loss by nursing mothers Breast-feeding mothers secrete calcium into their milk. Some of the
calcium may come from their bones, so mothers may lose bone mineral. Researchers measured the percent change in bone mineral content (BMC) of the spines of 47 randomly selected mothers during three months of breast-feeding. $^{26}$ The mean change in BMC was 3.587% and the standard deviation was 2.506%.
(a) Construct and interpret a 99% confidence interval to estimate the mean percent change in BMC in
the population.
(b) Based on your interval from (a), do these data give good evidence that on the average nursing
mothers lose bone mineral? Explain.

Problem number 52 which in a and we can see that the center for Noah were ah nonpregnant there seems to being in 100 grand center for the prescreening. Having was never pregnant has higher meaning and possible lying more to the right spread for the pristine. The group seems to be more than the spread for the not bringing group because it has higher standard deviation. And the with between the whiskers off the box plot is a more distribution. Appear to be a both rights killed because the Dominion which in the 19 the bus are the box block, appeared toe be more at on the left in the box are the parts of these air explanation Let's move toe 52 are being I expect it to be. Ah, I can. From beginning, you can see that export is even through political. Three point flying 87 X Bar two is equal to 1.309 or 91 It's one of the great toe toe Tribal 56 Word is two is equal toe 1.2 unmarried Ah, a 32 in one staples to 47 in cool is equal to 22. We can also know that it you note no hypothesis. Is that mean one? It would mean to on the good hypotheses is equal to and one a smaller van mean to Would it remind with the the sweetest? This is that this is justice statistics weaken severity is equal to a new one to admitting 3.587 ah, minus a new tour. Gino 0.3 09 or 91 over a square root, uh, to a point. My 0.61 uh, squared over on 47. But once we're right at to realign a 32 ah, squared over at 22. That's so 11 r b negative 8.49 Big reason. Freedom. It's a 1,000,000 off 47 minus one. Or to make it two minutes one And so it best most value is to make you one accordingly. On even is a probability off painting the video off the heaters or a value more extreme, be very in the number or in terrible. In the column titled Off they will be containing the event in Their Off Dignity or Freedom at 21 eso from the table and was given a really freedom equipment you want. We can see that. And the value is more than one. Well, trying on. And as we know, it really is it than or equal to significance level. They're not happen. They respected on. And you are. We know that be ready is smaller than a little time. So we can. But there's a formal high positions. So how answer finally will be here. Very sufficient evidence. And it goes upwards. McLean. That's I mean, parents we didn't see is a significant No. What? For the mothers. Her breasts. Okay. Before a question. Be good. Should see. Um so we can see that And its betterment And the lip if we imposes some. All right, treatment on the red. Words in order. Better to do. Ah, my responses. An observation, Baby Tries again, Man. Information your notes. Beings for me. Seen bay. All right, Germany. Just you there. They're Ah, vaginal. It's stunning. Okay, Figures in Yvette since the study. Uh huh. An experiment. We can't a conclude you should. And because this 30 might Hey, for instance, Brian ah, lurking earlier Also, we can see that working Manu. He's a very event. And that's and worth tens effects on ah relation relationship among and many of us among the very well study. Uh, not one? Uh huh. Simply referee. Then you mix. So if we want to prove privation random, an experiment is required, so I will answer Finally Would be would be No, this is an answer for quiches. A see through a question d We need to get clicks the in points So first we can get at the store. Ah, the storm from people it will be And it's really big with no point All it Oh, um So everything points and will be a new one with Camille one study Fine. It's 73 minus you to operate three on 9091 over is great routes. It's 1.5561 in squared over 47. Those weren't tow line A 32 They're over 20 between you two. Uh, it's not over. Is minus the store. Good. Like my wrist. So the federal energy project will be years before going eight. It's fine. Is one off in point. The other in point is the same equation, but ah the difference in that's the negative should be equal and should be plus, to get the other in point, which will be a still point burning for or think because he's of eight points for the degrees of freedom. Ah, for because of freedom. Ah 21 confidence in terrible decline right before I person.

This is problem number 73 from section 9.3. Uh, this problems talking about healthy bones and that they're the RD a recommended daily allowance of women there recommended you that women between ages of 18 and 24 get 1200 milligrams of calcium. Uh, however, they believe that this intake is actually less, uh, what really happens. So they did a study. They took a random sample of 36 women. And they put all that dad in the mini tab and you have the outputs there actually ran the test, um, for you in the second box. And then they ask us three questions So question letter a. Determine whether there's any out, Lars, and this is sort of a review from way earlier in this book. So if you remember, you have an outlier rule. The outlier rule is 1.5 times the Iike you are like you are that is Q three minus. Q. Once arrange Q three minus Q. One. So we're going to go 1090 0.5 minus 632 0.3, and I will get an I q. R of 458 0.2. Then find out my, uh, 1.5 times i q r rule. We're going to go 458.2. We're gonna take the Ike you are. We're gonna multiply that by 1.5 and I get 687.3 now, to find out if there's any out liars, we're gonna take that value. And we are going to subtract from Q one to get a low point. So anything below that mark is an outlier, and then we're gonna take the 687.3. We're going to add it to Q three and say, Hey, any value above that one is going to be an outlier. So let's start off here with q one is 632.3. We're gonna subtract out at 1.5 rule, so 687 0.3 and we end up with negative 55. Now they tell me my minimum value is 374. So there's no lo outlier, no lo outlier. And then we're going to do the same thing for the high end. We're gonna take you three and we're gonna add in that 1.5 rule and this is going to give me 1777.8. My maximum value here is 1425. You seat in a box there, so there's no hi out liars. There's no outliers here. So now moving on the part beat Interpret the P value in context. Now they tell me in that middle box at second box at the P value all the way. At the end, there is 0.0 It's approximately zero. So to interpret that what a P value means is, if the no hypothesis was true, if the no hypothesis is true, the probability of getting the sample result that we got is the P value. So here we go. If the true calcium intake for 18 to 24 year old women is 1200 milligram. So if that's true and we take a sample size 36 the probability of getting the result we got of 865.2 milligrams is approximately. The P value is approximately zero. So you could see my Senate, Sir. It's a true calcium intake of 18 and 25 year old women is 1200 milligrams. The probability of taking a sample size 36 from the population to get a sample result of 865.2 milligrams is approximately zero. This is one of the sentences that I have my students memorize, which is some blanks in here so they can fill stuff in, And then part see, do these daddy give convincing evidence to support the researchers? Sufficient suspicion carry out attacks. So we're gonna do a state plan. Do conclude here. So I'm going to do this a little quicker here. We've already done a couple of these. So state will perform a one sample T test again. Name of the test, given my awful value to see if there is convincing evidence of support that women have a lower calcium intake than the recommended daily allowance. So I'm sort of writing that Exactly how the questions asked I need to define my mu mu equals a true mean daily intake of calcium for women 18 to 24 year old Uh, my no hypothesis. And I forgot something on here. I need to put, um you in here? Mu equals 1200 milligrams. My alternate hypothesis Mu is less than 1200 milligrams. Okay, so our d a wants us have 1200. We believe there's less than 12 hundreds. That's my ace. Abo and H today plan three things random 36 women were randomly selected, 10% Rule 36 women is definitely less than 10% of all young women and large counts. Thank goodness we took a sample bigger and 30. So I can use the central limit theorem here and say that we can perform a T distribution. The do part and all these values are ah in those charts there. So we know that X bar, my mean, is 856.2 milligrams. I know My standard deviation is 306.7. My t stat value is massive negative 6.73 were over six standard deviations away, which gives me a P value of approximately zero conclusion again. Since the P value is approximately zero, which is less than 00.5 which equals Alfa, we reject the no hypothesis. There is convincing evidence towards the alternate. There is convincing evidence that 18 to 24 year old women consume less than 1200 milligrams of calcium per day

Hello. We would like to know how many adults have osteoporosis. So the first thing that we could dio is we can make it, Uh oh. Belle xnyr and or mean In this case, we're talking about young or no adults. It's going to be zero, and we're looking at people have osteoporosis. If they're native, 2.5 below that mean so we'll be here. So the amount of people that have osteoporosis will be too late. Um and we could actually break it up a little bit further to, let's say, a negative three here and this little red part right here, this will be 0.1%. And this blue part here is going to be 0.5 person, and if we were to add it up, will get 0.6%. So we know 0.6 percent of young adults have osteoporosis. Now in part B, we're going to talk about older adults when we still want to know they have osteoporosis. We can still draw a bell shaped curve. The only thing now is that he says, instead of mean, being zero, Armenia is going to be negative to, So he has shifted towards the left In order to figure out the number of people with super roses, we have to normalize this. So what is it normalizing? What? We're trying to find the probability that X is lesson native 2.5. Okay, well, we need to normalize it by that is first we need to to attract Why, that mean which in this case is going to be to But it's, ah, the media's attitudes. Who becomes plus, And they were gonna divide by the star integration. So standardization is going to be one. We're going to do it to both sides, so it's gonna be native 2.5 plus two, divided by one. When we simplify this, we're gonna get the probably Z is was in negative point by And have you close into your calculator, you're going to get point Jew 0.0.30 show about 30.85 percent a woman have. It's a process

So we're giving a table and it has 18 pieces of data in it. We are looking for the 95% confidence interval. We are, let's see, we were given the standard deviation and expire is the average. All I did was I added up all of those numbers and I divided by 18. So that's how we get expire. Now. The first thing we do according to our steps for 8.1 is find that and we're doing one minus the confidence interval and we're getting point oh five because its two tails, We are dividing it by two and when we look up that number on the table we get 1.96 and we're going to use that and step two and step to this is going to give us the range that we need for our confidence interval And ask the four million news every time and I feel that all in with the numbers I have up there, so the range I got was 860.54-1034.24. What does that mean? How do we interpret that data? We can be 95% confident that the mean calcium intake is between 860.54 and 1034.24.


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