Over this problem. You are the problem. Ask you, Thio Estimate. And I'm taking a confidence interval for the percentage change in price when 150 square foot bedroom is added to a house and you are supposed to estimate a an equation that has the log of housing price as a dependent variable. So let me just write that out really quick. This is your dependent variable, and then you're independent. Variables are square feet of the house and then the number of bedrooms in the house, and I'll just write out the estimates you'll get. So you'll run that regression. As the problem tells you to do with the data. And you should be, you should get thes uh, these coefficients and eventually standard errors from the estimation. So there's your intercept, and then this is the coefficient on square feet, and finally you have the coefficient on bedrooms, so both the coefficient for square feet and bedrooms air positive. That makes sense. Um, you know, basically means figure. The house is almost the more expensive. It like Leah's the price. So there we have the coefficients estimated, and then I'll just add in the standard heirs to which we also want to see in blue. So here's the standard error for the square foot variable, and it looks like it's a lot smaller than the coefficient. So the square foot variable seems to be very statistically significant, and the bedrooms variable does not look to be nearly as statistically significant. The standard error is actually little bit bigger than the coefficient that's we end up with. I'm just adding quickly the sample size as well, which is 88 and the R squared for us is 0.588 So these should be the results you get from the running the regression in your software. But this is now number. That part one is asking you to do exactly they're at. They're asking you, Thio, obtain a confidence interval or estimated survey estimating it, um, for the percentage change in price when a 150 square foot bedroom is added to the house. So what does that look like? First of all, just remember, we have this variable to square foot variable, and we have this bedrooms variable. And so what's an expression for adding ah, 150 square foot bedroom on so we could just call this data one. This is adding on 150 ft bedroom, so 150 square foot and then we have to multiply that by beta one. So 150 times the impact of a square foot, you can think of it that way and then plus a bedroom. So you're adding a bedroom and you're adding 150 square feet is a way to think about it. So you're adding a bedroom and you're also adding 115 square feet. So the state of one once we plug in the numbers that we got for beta one in 52 come out to the following numbers get 150 times beta one that we got, which is 10.379 and then plus 0.289 The notice, right? Super. Clearly they the one hat because I guess estimated it now. So I put a hat on. It equals 0.8 58 That's what you want to get for data one. All right. And that is basically saying that when you add 150 square foot bedroom That will increase the expected price of the house by about 8.6%. So from there we go on to part two, which is ask you to write beta to in terms of theta one and beta one and then to plug it into the original equation. So we've already written out an expression for theta one in terms of beta one and beta to So we just have toe rearranged a bit and that will just look like this. We will get beta two equals NATO one minus 150 beta one, and we can than just substitute into the original equation. So again, we have log of prices are dependent variable. Then you have our intercept beta, not plus beta. One times square feet. It's that hasn't changed. Then we have our substitution area here. So instead of beta two again, we're going to just put fatal one minus 150 beta one that all multiplied by bedrooms. And then, of course, add the at our air term at the end here. So that's what that looks like once you've substituted baited to Are you sorry? You've redefined beta too. And plugged it into the original equation, not just rewrite it a different ways that it's easier to estimate pretty simple rearranging here. Just isolate beta one out front, multiplied by square feet. Sorry, there. Square feet minus 150 bedrooms and can't forget beta. One times, bedrooms That will be our last variable at the end and our error term. So that is basically the final answer. Report to this is the substituted, the newly substituted, redefined original equation. So we have beta one as a parameter, and fada one in Beta two is no longer part of the picture. So Part three asked you to use part to obtain a standard error for data one, and to use it to construct a 95% confidence interval for that change in price. So the way that works, I'll start really quick. We want to get the standard error of NATO. One little hat there. To get that, we have to run this regression in again whatever program you like using. So have to run that regression. Um, better not plus beta, one times square feet times or sorry beta one times square feet minus 150 bedrooms plus data, one times, bedrooms plus our air. Jim. So run that regression that will get you a standard error on data one. And once you look at whatever the output is from your aggression, you should get something like the following. You should get the standard error of NATO. One should be 0.268 And remember, from part one, we already got our data. One hat estimate here so we can use that in conjunction with are standard error of theta one hat. And also look at what your software package says about the confidence interval, which for me, waas holidays a different color here, Confidence interval equals bracket. This was signifies. A range is from 0.3 to 6, 2.1 39 and then just put that in economic terms. This is saying that the confidence interval for the percent change in price when a 150 ft bedroom is added to a house is anywhere from a 3.3% increase in price to all the way up to a 13.9% increase in price. Little percentage signs there, so don't get confused. And that is then the problem