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At age 9 the average weight (21.3 kg) and the average height(124.5 cm) for both boys and girls are exactly the same. A randomsample of 9-year-olds yielded these res...

Question

At age 9 the average weight (21.3 kg) and the average height(124.5 cm) for both boys and girls are exactly the same. A randomsample of 9-year-olds yielded these results. Estimate the meandifference in height between boys and girls with 95% confidence.Does your interval support the given claim?BoysGirlsSample Size6050Mean Height, cm123.5126.2Population Variance98120

At age 9 the average weight (21.3 kg) and the average height (124.5 cm) for both boys and girls are exactly the same. A random sample of 9-year-olds yielded these results. Estimate the mean difference in height between boys and girls with 95% confidence. Does your interval support the given claim? Boys Girls Sample Size 60 50 Mean Height, cm 123.5 126.2 Population Variance 98 120



Answers

Heights of 9 -Year-Olds At age 9 the average weight $(21.3 \mathrm{kg})$ and the average height $(124.5 \mathrm{cm})$ for both boys and girls are exactly the same. A random sample of 9 -year-olds yielded these results. At $\alpha=0.05,$ do the data support the given claim that there is a difference in heights? $$ \begin{array}{lcc}{} & {\text { Boys }} & {\text { Girls }} \\ \hline \text { Sample size } & {60} & {50} \\ {\text { Mean height, } \mathrm{cm}} & {123.5} & {126.2} \\ {\text { Population variance }} & {98} & {120}\end{array} $$

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.

Here in this problem, we have to find the main deviation about me by using the given data. So in this table we have given height in centimeter number of boys, that is a five and and I have found the mid value that is exciting. Now, in order to find the mean deviation about me, first of all, we will find if I excite, So here we have 102, 9 is 900, 110 into 13 is 1430. 120 into 26 is 30. in 230 is 3900, 114 to 12 is 1680 And 115 to 10 years, 1500. Now we will find the mean, that is Mean as equals two, one upon N sigma. If I exciting. Well, I goes from one to n by substituting the known values, we get Men, which is equals two, one upon 100 into 12,530 It would be equals two, 125.3. Now we will find the absolute value of deviation from the main. So here we have XI -125.3 since we are finding the absolute value, so we will ignore the negative sign. So here we have 100 -125.3 is 25.5 100 -125.3 years, 25.3, 110 -125.3 is 15.3, 120 -125.3 is 5.3, 130 -125.3 is 4.7, 140 -125.3 is 14.7, 150 -125.3 is 24.7. Now, we will find a fly into absolute value of deviation from the mean, it would be 925.3, which is 227.7 pertaining to 15.3 is 198.9 26 in two, is 137.8, 13 to 4.7 is 141, 12 into 14.7 is 176.4, 10 into 24.7 is 247 and this emission is 1128.8. Now the main deviation about mean is equals two Van upon n sigma phi In two XI -125.3. Well, I goes from one to and by substituting the known values we get mean deviation about mean is equal to one upon 100 into 1,128.8, Which is equals to 11.28. Hence, we get me innovation about me As equals to 11.28.

All right, this question. We're talking about the average height of the boys and the girls. So you can see the data from this flat sample tells is that, uh, the average boys height so this is given eyes 1 37 centimeter at age nine. And similarly, there stole, um, boys Height. I, uh, age 11 is going to be 1 50 ones in dignitary collect on again. Additionally, they're also telling that average head off 11 year old children. So he said that average height off all live in your old children is approximately given as 1 15 centimeters itself right now for the foot force. But you're supposed to check. Uh, I mean, if the girls and boys are off the same hydro no one an average at age 11. So here we can see that, boys approximate average boys average at age 11. Deaf centered. There's 1 51 scenting jos. Correct. And, uh so therefore, beginning for that, girls must also average 1 50 ones and Dean eager at age 11 because they have given us the second bottle Here. You see this in at the average height off all your 11 year old is 1 50 ones and emitters. So because if girls were stolen and 1 51 and they would have not mentioned overall average here and if girls the shorter than water lavage would have been actually less than this 1 15 months endangers. So therefore we conclude your that the goods and the boys are of the same height, which is Ah, 1 50 ones and demeanor at age 11. So you were done with the body. Now we're going to the party. We were now in partly supposed to guess, the average height of 10 year old boys. So they have to find the average height off 10 year old boys. So this is what we're supposed to find now. The data has oldest that in a two year period. So if you talk about a two year be idiot. Okay, then, uh, safe from age nine through 11. Okay, The average boy grows approximately 14 sending doors so we can estimate that from age nine to h. Then he's grown up. Broke seven. Sending does so. I'm writing here that therefore from sea around age 9 to 10 years grown by, see how for food in just seven centimeters and there's the best guest for the height off. 10 year old boy in this case will be approximately the calculation and presenting here. Now if I see that if it's a two year old root beer, then that is 14. Sending turns right. So therefore, if it's a one year growth video, everything earlier and this is half or four beans or seven centimeters now, the way by which I mean the guy's like of the height at age 10 someone to write a letter that hide at age. Then it can be to do by the phone like that. It is high it at age nine, correct and bless the average when you're good. All right, so the height at age nine has been told as 1 37 and the average of January's 70. If I add them, the answer is 1 44 sending George. So that is why we guess that our best guess for the height off a day in your old will be 1 44 Sending it does here

We want to construct in 95% confidence interval away. Okay, then we have a random sample off 860 birds in New York state. 860 words 860 bucks. This is my sample size and this sample size included 4 26 voice 4 26 voice. So if I use my calculator, what is the proportion for 26? Divided by 860 0.4953 This is 0.4953 This is my peak cap. This value over here is my sample proportion. Now I want to construct a 95% confidence interval. So how do I do that? First, I need to find the margin of error. What is going to be the margin of error? It is going to be easy, Alfa, by to where my Alfa by two is going to be 0.0 to fight. And for 0.25 my Z alphabet is 1.96 So this is 1.96 multiplied by route over. This is going to be multiplied by route over P cap. That is 0.495 three. Okay, Multiplied by Que ca, which is going to be zero point 50470.5 047 Okay, divided by en And is what? Sample size, which is? 860. So if I can play this, this will be rude. Off 0.4953 multiplied by zero point 5047 Divided by 8 16. Divided by 860.17 This turns out to be 0.17 This is my error margin of error. What is my peak cap? 0.4953 I have my p cap. I have my e. So what is going to be my confidence interval? My confidence interval is going to be zero point. Or let me just like this pick up minus e less than p which is less than p cap less heat. And this value actually turns out to be 0.46 to 0.46 to less than three less than 0.5 to 9. 0.5 to 9. This is my confidence interval at 95% confidence. Now it is actually believed that among all the birds, the proportion of boys is 0.512 Now, do these samples provide strong evidence against the belief? No, we cannot say so. Why? Because 0.512 is actually included in this interval. So we cannot say that we have strong evidence that this belief is wrong or to suggest anything otherwise.


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