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We used the data in MEAP93 for Example $2.12 .$ Now we want to explore the relationship between themath pass rate $(m a t h l 0)$ and spending per student (expend).$\begin{array}{l}{\text { (i) Do you think each additional dollar spent has the same effect on the pass rate, or does a dimin- }} \\ {\text { ishing effect seem more appropriate? Explain. }} \\ {\text { (ii) In the population model }}\end{array}$$$m a t h I 0=\beta_{0}+\beta_{1} \log (e x p e n d)+u$$argue that $\beta_{1} / 10$ is the percentage point change in $m a t h I O$ given a 10$\%$ increase in expend.$\begin{array}{l}{\text { (iii) Use the data in MEAP9 to estimate the model from part (ii). Report the estimated equation in }} \\ {\text { the usual way, including the sample size and } R \text { -squared. }}\end{array}$$\begin{array}{l}{\text { (iv) How big is the estimated spending effect? Namely, if spending increases by } 10 \%, \text { what is the }} \\ {\text { estimated percentage point increase in } m a t h 10 ?} \\ {\text { (v) One might worry that regression analysis can produce fitted values for math } 10 \text { that are greater }} \\ {\text { than } 100 . \text { Why is this not much of a worry in this data set? }}\end{array}$

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(a)$\\$)$\\$In case of a log-log model (as in this case), the slope coefficient $\left(\beta_{1}=0.312\right)$ measures theelasticity. That is, it measures the percentage change in the price level for a given percentage)$\\$change in the distance from the garbage incinerator.)$\\$The sign of the slope coefficient is "positive" exactly like it is and should be expected.)$\\$The coefficient on log (dist) is interpreted as one percent increase in the distance from garbage incinerator is associated with $0.312 \%$ increase in the price level and vice-versa.)$\\$If living closer to an incinerator depresses housing prices, then being farther away increases housing prices.)$\\$(b))If the city chose to locate the incinerator in an area away from more expensive neighborhoods, then log (dist) is positively correlated with housing price.)$\\$

The strength of the correlation is not quite strong as it is close to 0(0.162) . If the ${ }^{2} R^{2}$ ' is close to '1', then the strength is said to be strong.)$\\$

The estimators are unbiased and the elasticity is a good representative of the true elasticity of price with respect to distance.)$\\$

The casual or ceteris paribas effect of distance of price means that it is better to have a home)$\\$farther away from the incinerator)$\\$.(c)When some of the variables like log (land), log (area), and log (rooms) are added to the regression model, the effects are much smaller and insignificant as a result.)$\\$All these factors affect the housing price and might be correlated with the distance from the)$\\$incinerator.)$\\$

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moving on to the six computer exercise. Now we use the data set called me a P 93 1 explored the relationship between the math pastorate, which hold Mass 10 and the data said, and spending per student. In the first question, we need to answer, Ah, theoretical question. We're being asked if we think that each additional dollar dollar spend has the same effect on the past rate, or does he have a diminishing effect? Which one seems more appropriate? Well, this thing about it for a second, let's consider two different schools, one that is low spending schools. Another one is high spending schools. I think we will agree that it seems plausible that another dollar of spending has a larger effect for low spending School's done for high spending schools, all right, because that low spending schools, more money can go towards purchasing more boats, computer and for hiring better qualified teachers. Now, on the other hand of high levels of spending, would expend if any effect because the high spending school's already have high quality teachers have nice facilities, planet books and so on. So I think that seems very plausible. Miss Hypothesis and put it in a more mathematical way. We would say that, um, the function that maps the function describes the relationship between expansion receding and the math. Past rate has a con cave shape. Right? And what does this mean? This means that concave a function has the property that the partial very very, But each point, as we increase that X variable decreases, right. We have diminishing returns, which is captured by diminishing, um, value of the partial derivative. This means remember, the partial derivative or in this case, is by viruses, totally derivative. It is the slope of the tangent line of it. So if we are here in the beginning, the attendant, the slope is very, very high. But as we move, the slope gets low. Ah, smaller and smaller. Here, you could see it slower, smaller, And at some point where that line reaches a plateau, this love will be zero, right. There's no more expensive dream. Imagine the, uh, the the highest levels of spending in the world. If we increased by $1 it will make no difference. Whereas it's the same thing, right? If you give $1 to Bill Gates, it will make virtually no difference to him. If you give $1 to a four person, it will make difference for real. All right, now, moving on the part too. Um, in the population model math tentacles interested. Please be the one lug expenditure plus Ah, with servants. Erm argue that be the one by the by 10 is a percentage point. Change in math and given a 10% increase in expect. Well, that's not very difficult to do, right? If we take changes as usual, as we do the Delta operator here we get the difference in the math. Ah, scoring meth and score equals to be the one times that lug difference of the expenditure. Now we know that that love different is the approximate ah percent of change. So if we put, um, if we substitute in the the expected percent changes love math. It is Ryan Delta explained, is equal to 10. Were we together that the difference in math grade is equal to be the one divided by 10 exactly as we're being asked to do. This is not This is not any difficult thing mentioned. You just manipulate the equation in a straightforward way and substitute the value of 10 that will be given into the equation. All right, now, in part scene, we use the data. Said, uh, M e a p 93 twist made the model that we just is this tribe from Part two and report the estimate equation in the usual way. Well, let's hit described first to see if we indeed have the right data said Right 104 108 observations. 1800 lying variables and we're interesting. Expand all right, expanded, appreciating dollars and meth. 10% is of Susan's passing the M E a p meth. And here we need the, uh, we need the luck transformation of expands like Alec spent. So what we need to do, right? It's a regression off math. 10 dependent variable with a constant with intercept seven that we don't need to write down. And what's the name of the other one? L expend rights. I'll expands. Here it is. Okay, the number of observation isn't this correct? 408 Ah, the model does passed a simple join f test where the coefficients, meaning that we have sufficient evidence to say that the model is but better than just a simple, interested model. The escort is not too high, but not negligee Boys around. The variation in Lug of expenditure explains. Around 3% of the percent is a student passing the math exam again. You know it's not a significant are square, but it's not series on like an invisible. And most importantly, the estimated coefficients are statistically significant. Maybe maybe we don't see ah, extremely high tea value. That would make us happier. But we basically statistical significant least of the 1% level. The coefficient for the interested term is minus 16.34. And what we're interested in the coefficient for legs centuries is 11.16 and I've written down the estimate equation right here. Seven point. It seems like expand observation in the square. I know in part for how big is the estimated spending effect, namely, spending increases by 10%. Like we said before in question, too. Um, what is estimated percentage? Porn increase in math, Stan. All right. As we said before, we said that the this difference will be able to be the one divided by 10. Here, we're gonna use the be the hat one that we have 11.16. So if expend increased by 10% our estimated approximate percent huh percentage approximate increase of the mast and variable will be 1.2% points. Why? Because it's 11 points 16 divided by 10 is one point around 1.1. Again, this is not a huge effect, but it is not trivial for low spending schools. Right? A 10% increase in spending, likely a fairly small dollar amount in absolute in ah levels, right? So again, this is not trivial. And we shouldn't just discard this analysis because we think this is a small number now in Part five, um, one might worry that regression analysis conduce feeding values for methane that a greater than 100 whys. It's not much of a worry in this data set. Indeed, as we've seen previous question, this is definitely an issue. But in this case, let's see what's going on. Let's just summarize, um, math. 10. It's weird to see what are the have, ah, simple picture of what's going on. Descriptive analysis. We're seeing means around 24 the maximum Here's the thing. The maximum level is 66.7 and this is not really close to 100 right? If that was close to 100 then there would be a risk that we would produce scoring more than 100. And indeed, this is the key to understand why we don't worry here. And indeed, let's do something. Uh, I'm gonna use the command. Predict Predict produces the fated values from the regression. So the math 10 hat that we saw before Let's call these very well fitted. There's just no. One variable, right? So the new variable fitted here would have predicted values. Eyes already. Fine. Okay, did before. That's fine. So if we summarize this fitted, we'll see that indeed, the maximum estimated we gives up around 30.2. And indeed, this proves that where there's nothing to worry about in this day to say

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