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Why do we write the estimate b1 as Zk;Y ; where(X-X) ki = Z( -X?a. It is easier to prove the biasedness and the minimum variance for b1 using the kj properties: b U...

Question

Why do we write the estimate b1 as Zk;Y ; where(X-X) ki = Z( -X?a. It is easier to prove the biasedness and the minimum variance for b1 using the kj properties: b Use kj properties to prove various properties of the sampling distributions of b1. Use k; to write b1 as a linear combination of Xid. All of the above

Why do we write the estimate b1 as Zk;Y ; where (X-X) ki = Z( -X? a. It is easier to prove the biasedness and the minimum variance for b1 using the kj properties: b Use kj properties to prove various properties of the sampling distributions of b1. Use k; to write b1 as a linear combination of Xi d. All of the above



Answers

(a) Recalling the definition of $\sigma^{2}$ for a single rv $X,$ write a formula that would be appropriate for computing the variance of a function $h(X, Y)$ of two random variables. $[$ Hint: Remember
that variance is iust a special expected value.
(b) Use this formula to compute the variance of the recorded score $h(X, Y)[=\max (X, Y)]$ in part (b) of Exercise $24 .$

So using the hint for this problem we have that variance of a experts be is equal to the expectation of a experts. Be minus the expectation of experts. Be inside Prince he's squared, then close bracket. Now this is equal to the expectation of a X plus be minus eight times muse of exposed be and this quantity squared, which is equal to the expectation of now that we have to be here and they'll be here, we can cancel them. So we have the expectation of a X minus a mule X squared, which is equal to the expectation off a squared times X minus mu X squared and this can be written us This could be redness, some of a square times x minus mu squared times pubic ce. And this is by definition, where ex lives in X. So now we can take out the A squared out of the summation because they escort is a constant with respect of X variable in Catholics. So we have a squared times a summation of X minus mu squared times p of X, where again excellent. The Knicks. So this is equal to a squared times Myriads of ex capitalists because Catholics is a random variable here and we're taking the variance of that while in this definition, X is a realization of the random variable where we're summing over the values of X dorky sex in capital X.

All right here on this problem, we're dealing with properties of expected value and variants. Now we are given the expected value of Y is new, the variance of why is sigma squared? And then on a new industry variable X. Does he go to Y plus one? And we're going to answer some questions here about ads question A asked us if we would expect the expected value that to be larger than or smaller than that for why we would expect it to be larger and think about why this is you took every element of why And then added 1 to it. If you take every element of wine and add one to it, that should increase what the expected value is. And so naturally we're expected to be larger. Uh huh. Now and be we want to find the expected value of vets. Well that's that's why plus one. So this means this is E of Y plus one. You have y plus one Izzy of Y plus be of one. The expected value of a constant is just that constant. And so you have one is just one and then any of Y. Is what was given to be mu. And so this means that you bet Is a problem. You plus one. Now on C. And D. We're gonna be doing with the variance on C. We want to state whether we think the variance of arts is going to be larger than smaller than or equal to sigma square, which is the variance of Why? Now we would expect it to be the same variance. Mhm. Is how spread apart our data is but adding one does not change that just by adding one to each of your data sets. That does not change how spread apart your data is. And so we were expected to be the same. And so let's show that here on D the variance of acts 0 to the variance of Y Plus one. Okay, I just received your like this one which is just equal To the variance of why. And this is from theorem 36 various wise sigma squared. So the variance here is the exact same and that's what we wanted to show.

In question. 13.3. We're going to be using some by various data to walk of the co variants, the standard deviation off the X values in the white, my leaves and the coefficient of clinical relation. And he had the formulas that we're going to be using. And these are the columns that we'll have to create. So let's begin. We need a column for the sums, and also we need a roof for sums and a rueful means. So the first thing we need to get other means for X I can walk out the means of X by getting formula of rich a rich all the wonders of X here. And you think, why those two will give us X bomb and why bar, respectively. Now we can start the Fasts column for X minus X bar. So it's going to be equal to X minus the expert, which isn't 17 biscuits. No, through we also need why minesweeper, which equals why minus y bomb. Which is that and we before me, law. True. Next, we need the product off the deviations that is, uh, the new miniter for a collision for co variance in this case is going to be No, they get 50 times true. Now we'll need the some off This column to C equals some like all the Bradleys in that quote to Oh, you met her for the formula for co variance is going to be equal 300 divided by n minus one and we only have six values, so n minus one will be five. So in this case, the co variance He's 60 in part b of the question. We're going to get the standard deviation off the six values of X six miles of Why so the formula for standard division is given, So we just need to square the deviations for X so x minus X bar square. So take the X minus X box and raised it to the part off to Formula One through. Then we need to get this sound off this God divisions. And now we can walk out the standard division back so it's equal to the square. It Oh, some 8400 divided by n minus one, which is five. That gives us 40. I don't need synthesis. Tentative aviation, Then the next thing we have is a standard division for off the six y valleys. So we're going to get to feel this column. Why minutes? Wife asked. Glad so we're going to scrap this degrees. That means you re that part, too. We need to get a total She's 2200. So standard deviation House the 65 equals this square it divided by five, which is ended. This will be I'm it in Patsy of the caution going to calculate the linear correlation coefficient R are going to use this formula, so it's good to be equal to the core variance, which 60 divided by productive, off center depredations of Fuki many times and it's you look zero, 69 78632 In the last part of the question, we're going to be comparing this results with the results from example 13.1. It's touch with the values religious of the values that we hade. In example, 13.1 car two and 13 and 56 and 38 and to its own. And the pattern here is that the data, all the values in the details, have been multiplied by 10 su in our case X by 70. And why bomb? He started, in example, that in 0.1 x by seven, while y bar is three. So also, the means are such that one is 10 times the other king. So when we compare the co variance, the variance for it did huh is zero point six. But in our data, the compliance is 60. So you notice that the co variance is 10 times the co variance off the data. In example, one. Now, when you check through to see the the variance off the deter if bodies of X, the violence off the X values is four point 099 in our example and in the question here we have 40 0.99 So that tells us that the variance was the standard deviation is 10 times the standard division. In example, one also in example, one The standard deviation for why is 2.98 any now question the sample division is 10 times as much. Lastly, we can compare the linear correlation coefficient. In a first example it is 0.7 and in our example it is 0.7 as well, so you'll notice that all the values will differ. The violence that differ are the means expert. And why, Bob? Also the Cove Arians, the standard deviations All of them are different. But the two sets of data have the same Lena correlation coefficients.

So here we're giving the sample data. 23 17, 15, 30 and 25 part A. We just asked to compute the range. So in this case, you just want to take the highest value minus the lowest values. This would be 30 minus 15. Well, give us a range of 15 and then in part B. Want to verify that Sigma X is equal to 110 and Sigma X squared is 2000 568. So said my ex is going to be 15 17 plus 23 plus 25 30 which does equal 110 Sigma X squared is just 15 squared plus 17 squared. He's got one. Yeah, which does equal 2568 And then for part C, we want to use the results of part B and the appropriate computation formulas to compute the sample variance squared in sample standard deviation s. So the simple variants R squared is just equals who? Some of X squared minus Sigma X squared over n divided by n minus one. Wherein is just the number in our data set. So this would be equal to since we've just computed X squared, the son of X squared, I guess 110 squared over five mhm we were for this will give us s squared is equal to 37 and then to compute the sample standard deviation us is just the square root of the simple variants. This will give us a sample standard deviation of 6.8 and then in part D. We want to use the defining formulas to compute the sample variance s squared and sample standard deviation us. So in this case, squared is just equal to the sum X minus X bar squared over n minus one. And so here we're going to have to compute the mean, um so we can plug it in because that's what X Bar is. So we can go ahead and do that. The formula for the mean is the sum of each individual. Data points over the number of data points in the set. And so let's just go ahead and compute the mean really quick. So it'll be 23 plus 17 plus 15 plus 30 plus 25 divided by five. So I mean will be 22. And so the top of this function, um, the first term would just be 23 minus 22 square plus 17 minus 22 squared and so on, and plug in those values will give us a value for us squared of 37. And then the sample standard deviation s is just the square root of the sample variance, which is 6.8 Yeah, and then an e. Suppose the given data comprise the entire population of all values. We want to compute the population variance sigma squared and the population standard deviation sigma. So signal squared is the population variance and it's just the sun of X minus X bar squared over the number of data points in our set. So similar to what we didn't part D. It's just x minus X bar, but the some So I'm taking each data point. But this time it will be over five. This will give us a value of Sigma squared of 29.6 and then the popular the Yeah, the population standard deviation sigma is just the square root signal squared. This will give us a sigma value of 5.44 Yeah,


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