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Quation 2 (Jpoints] large e school district is reevaluating its leachers' claries For , such evaluation, it is using its recorded data in the last 20 years on ...

Question

Quation 2 (Jpoints] large e school district is reevaluating its leachers' claries For , such evaluation, it is using its recorded data in the last 20 years on tcachcns salarics (Y) and ycars of experience (X) Below isthe Excel output of the relationship between teachers' salaries and years of experience: SumMARY OuTPUTRegression Stotutlcs Multple 0,9629-4] Square 0.9272617 Adjusted square 0,9217207 Standard Etror 4123 8137 OpsentionsANOVASoconce 390220261} ~SE+09 1794025 1.093S9E-44 30

Quation 2 (Jpoints] large e school district is reevaluating its leachers' claries For , such evaluation, it is using its recorded data in the last 20 years on tcachcns salarics (Y) and ycars of experience (X) Below is the Excel output of the relationship between teachers' salaries and years of experience: SumMARY OuTPUT Regression Stotutlcs Multple 0,9629-4] Square 0.9272617 Adjusted square 0,9217207 Standard Etror 4123 8137 Opsentions ANOVA Soconce 390220261} ~SE+09 1794025 1.093S9E-44 3061051171 11005840 403401450 Regresston Residual Jotr cosLurau Standard {rror (stot VO QuLOS 3 Ta 695369 1JE-08 iAd Wano 4D L0Jt-I Lot 9S Uppen9s thon 070s| 1/0575716 oa 47Y 7197 uneceDi Ve Wlt the preclicthol (qju.ti(u? Husedl on the sugnutic.Mc e e thle modol Kood ptelicUth apuallon? MICCUE 01 tic Varuon testc hrs eplau Ub Vcan 0 enticin4



Answers

Teachers' Salaries The mean salaries $S$ (in thousands of dollars) of public school classroom
teachers in the United States from 2000 through 2011 are shown in the table.
A model that approximates these data is given by
$$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$
where $t$ represents the year, with $t=0$ corresponding to $2000 . \quad$ (Source: Educational Research Service, Arlington, VA)
(a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain.
(c) Use the model to predict when the salary for classroom teachers will exceed $\$ 60,000 .$
(d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

Part one. The smallest number of schools is one you may find that 271 out of 537 districts have only one elementary school in the sample. Yeah, the largest number of schools in the district is 162. Yeah, the average number of schools is about 22.3 mhm or to the old. L s estimate of beta B s is about minus 0.516 and the usual s standard payroll is 0.11 for three. The standard barrel on B s robust. You within district cluster correlation and heterocyclic elasticity is about point 25 three. This gives a T statistic on Bs of minus two point oh four, so T statistic has reduced, and now the variable bs is only marginally significant. At four, you may recall from computer exercise 12 in chapter nine, the coefficient on B s is very sensitive to the inclusion of the observation with the highest PS. When the four observations with Bs greater than 40.5 are dropped, our estimate on Bs becomes minus point 186 with a robust standard error of 0.273 and the T statistic, which is the ratio of the estimate data on the standard. Errol is minus 0.68 and given this t start, we are unable to reject the non hypothesis that beta bs is zero. In other words, there is little evidence of the trade off between salary and benefit. Thank you. First five, when we use all the observations, the fixed effects estimate of beta bs is minus one point oh five mhm. And the standard error is point 09 six. And this value is very close to the theoretical value of minus one. Yeah, and for sure, the 95% confidence interval contains minus one. If we drop the four observations with B s greater than 40.5, the fixed effects estimate of beta Bs becomes minus 0.5 chu three and the standard error is point 147 And now we can reject the null hypothesis that beta bs is zero. So, in other words, there exist evidence that there is a trade off between benefits and salary. However, you may find that the 95% confidence interval construct from mhm this beta had value and the standard error does not contain minus one. This is equivalent to saying we reject the null hypothesis that Beta had beta B s equals minus one. Okay against the two sided alternative. So again, there is little evidence that the trade off between benefits and salary is 1 to 1. Part six. We get such a result in part five and four because we have the district fixed effects. The district effect captures the fact that some districts pay higher salaries and higher benefits for reasons not captured by our assistant controls. And once we control for the systematic differences across districts, we find a trade off. The fixed effects estimate is based on the variation in salary benefits packages across schools within districts. Okay, One reason different compensation packages may exist within districts is because of different teacher age distributions with red within districts. Yeah,

So this is another problem of statistical inference here. This is one of the longer problems in the in the chapter. There's six parts to it, so we'll get rolling part one. So part one asked you to run the simple regression of the log of average salary. So this is what log of average salary looks like. Ah, running that on the variable B s, which is the ratio of average benefits to average salary by school. So you're gonna have an intercept term first, I'll just write out the estimated coefficients in the standard errors to start off, um, to expedite the process a little bit. So there's the intercept. And then here is our coefficient on de B s variable, so bs. So it's negative. Note that it's negative. So that's just saying that an increase in the ratio of average benefits to average salary. So, um, if average benefits are uh huh, closer to average salary, then the log of average tally will be going down the dependent variable and home. You have a standard error here just to make sure it looks like it's going to be a statistically significant coefficient. So just from this this result here, we can answer the first part of part one, which is is the estimated slopes statistically different from zero. And given that that T statistic here, if you look at it and the results the T statistic equals negative 5.31 So we know this coefficient is statistically different from zero. So we know that this ratio of average benefits to average salary has a non zero effect, a negative effect on average salary our association with average salary. Second part, though, is as asks if this relationship is different than from negative one. So for this, you just have to change the T statistic here. So instead of our normal T statistic, we have toe alter the formula and we have Thio, um, subtract from our estimated coefficient, have to subtract negative one, right? That's the, uh, alternative data hat, um or are that is our alternative Beta, um are alternative B s coefficient in the population that we have prophesized might be true. Some sort of zero. We're now saying it might be negative one. Um, that's the only difference in our T statistic. We have to divide this by our standard era that we got from estimation. So once you do this calculation, you should get a statistic of 1.33 And that's associate ID with a P value of 0.92 That's the end of the road for this first part, for part one here. So it's sneaks under the 10% level of significance, but it does not meet the 5% or 1% level of significance. So to answer Part one, we can say again here, that's the estimated slope is statistically different. Check mark. It is statistically different from zero, but this tells us that there's, um, we're not quite sure. So at the 10% level, yes, it is statistically different from negative one. But at the 5% level and the 1% level, we would say that it might not be statistically different from negative one. That's part one part two. We have to add a couple new variables to the regression, Um, and you want to estimate the the following equation in this time all right, out the whole ah, the Betas and all that good stuff just to make sure we're on the same page here. So just like the first part. We're going to have that BS variable, That ratio of average benefits, average salary and then I'm going to add two more variables. The first one is, uh, the log of total enrollment school. So another variable that might affect average salary, right. So including that. And then the second one we're including is the log of the ah, the staff. But this is the kind of the the ratio of staff to students. So it's the staff per 100 students taking the log of that. So this is our new estimation regression here. So we're just throwing in a couple more variables here to control for other factors that we feel might influence salary at the school. Mainly or specifically, I should say, the number of students or enrollment and also the ratio of staff 3000 students. So we want to find out what happens to the coefficient on on B s here. So let's write out. Let's read out what happens here. And also I'll just go right ahead. So, uh, let's go over here. So it's a you should get the following estimation for beta one hat. Niko Fish on. Yes, still negative. And 0.605 is the new coefficient. Standard error of point 109 And let's just compare that. So the first part, you're the first problem. We had a coalition of 0.8 or negative 0.8, I should say. And here is negative 0.6 approximately. So we can say that this estimated coefficient on the BS variable has gotten smaller. After adding our tomb or explanatory variables and what you can do for the next part, they ask you, how does the, um, situation compared to table 4.1? So you have to go endured textbook Look, a table 4.1 in there. And just look at how this compares to what they did. Their results there book. And what you should find is that, um, compared thio the table 4.1 results. This is what we're doing here. Compared to the table 4.1 results the coefficient. So this negative 0.65 here that we got for beta one hat that is exactly the same coefficient as the coefficient in table 4.1. Um, so let's say same beta, one hat. So that's the first part of that. But even it was the same beta one hat. The standard error we have is much smaller, so that standard error is smaller. That's what you should get out of that other part of part two here. So after making those observations were done with Part two and we can go ahead and move on to three, which is just just answering or explaining this part here. So explaining. Even though the are beta had coefficient on the BS variable is the same as the one in table 4.1. Why is the standard error smaller? And this has to do with, um, error variance and multi kalani aren t when we think about including these, uh, enrollment and staff variables and what you want to think is you want to think about the following process that happened. So remember we he added two new variables, right? So we added the log of total enrollment and we added the log of the ratio of staff to students. And what do you think about, um, adding these two variables that they might be, um, somewhat correlated, But you can think that these air probably un correlated with that Bs variable, so let's think about that. So the BS variable, remember, is a ratio of, um, average benefits to average salary. What's the ratio? Quantity? And you would think that if the enrollment of the school increases or the ratio of staff to students increases that this ratio wouldn't be correlated at all with those amounts, we have no real reason to believe that Bs variable be correlated all with changes in enrollment or the ratio of staff to students. So since these two variables are uncorrelated with RBS variable, that just means that the precision or, in other words, the standard error of the beta one had for the B s estimates increases the precision precision increases. Which means the standard error decreases the camp. So that going back just quickly, you cap reason the likely reason why our standard error is smaller. Then the table 4.1 standard error, even though the beta one hat is the same, is because we when we added two new variables, they're likely uncorrelated. Or maybe you know, very slightly correlated with RBS variable. So the precision of our beta one hat increases. Um, check the cross that the precision of beta one had increases, Which another way of saying that the standard error decreases. Sorry about that. This is a better way of saying it. And this is going back to their hint. They give you the problem. They ask what happened to the air variance versus multi clinic charity and think about if two variables are likely, not Kalin e er the precision on an existing coefficient would go up. But let's say if we thought that the log of enrollment in log of staff were highly correlated with her Bs coefficient, we would actually expect the standard error to increase. So that's part three. The ship part three There, part four. Ask you How come the coefficient on the log of staff here, this variable How come this coefficient is negative and is it large in magnitude? So all this first right out the the beta hat that I got after running this progression and like they said, you get a I did get a negative number 0.714 So think about why this might be negative. So we're saying that with a higher student to staff ratio so higher, if you have a higher student to staff ratio. Then the, um, average salary goes down. So lower average salary. So why is that? Why are we seeing that in the regression? What reason could be that either when you add additional staff to the school. So let me write this out. So maybe additional staff, um, command lower salaries. So maybe adding more staff, these newer staff are compensated at a lower rate than the staff that were already existing at the school. I was right that as additional staff command lower salaries, that's one possibility which would just shift that average salary distribution downward or sorry. Searched his shift. The average salary downward. Yeah. Additional staff command, lower salaries when you add them. Well, that's one possibility. Another simpler possibility is that the as you, ADM or staff for the given students, um, you can just think of maybe the total available salary compensation is split into more ways. That's kind of a simpler answer. It could be a combination of the two, um, but that those would be two reasons to think of why that coefficient is negative. And the second part they asked, is it large in magnitude? And the answer for this is yes, and it's actually very large. Um, and specifically, let me get this in red. Uh, so an increase. Sorry, I'll backtrack a bed. So a 1% increase in the staff to student ratio is associated with a 0.7% decrease in average salary. So that's a very big effect. You know, that's it's getting close to a 1 to 1 1 to 1 match or a 1 to 1 association there. So this coefficient is a very large effect. So that increase in the step to student ratio is associated with a pretty big decrease in average salary for teachers and school on average. All right, that brings that part four Part five. We have to add another variable here on dial just for, um this for simplicity. I'll just say that we have Thio run the following regression, um, again, Same dependent, dependent variable. The log of average salary here. And I'll just condense what we had from the regression in part two here. So all the variables into I wanna put in the B s and the log of enrollment and log of staff all those variables, but then add to that are lunch variable here, Caesar. Our new regression for part five. So and and lunch Lunch. Just to be clear, is the percent of students eligible for free or reduced price lunch at the school. It's kind of a measure of, uh, the students in poverty. Okay. And it is asking you, holding other factors fixed our teachers being compensated for teaching students from disadvantaged backgrounds and explain. So look at that. We are interested in this beta four here. So this state of four would tell us that as he percent of students in a school eligible for free lunch increases, how does average salary change? So if teachers air being compensated for teaching students with disabilities, the pictures are being compensated for teaching students from disadvantaged backgrounds. We would want to ask if beta four hat is positive. That would give us, uh, evidence that teachers are would be compensated for teaching students from disadvantaged backgrounds. So what do we What do we actually get here? So you should get that beta four had equals Negative point. Oh, 76 And with a standard hair of 000! 16 So there we go. It's the coefficient looks to be negative and statistically significant here so that the T value equals negative 4.69 So this statistically significant at the 1% level or lower. So it looks like, given this coefficient that we can say no that teachers air not being compensated for teaching students from disadvantaged backgrounds. Um, specifically. So the way this interpretation would go is that of 1% increase in, um, in percent of students eligible for lunch. So 1% increase in the percent of students eligible for free. Lunch is associated with a 0.76% decrease in average salary. So teachers are actually being, uh, it is It is, uh, not not as good for them even teaching at schools with a greater percentage students eligible for free lunch. So this kind of a strange interpretation, and it's not we're how big this impact is. Another way that I calculated this out being is that teachers at a school where um, or 10% of students are eligible for free lunch. So teachers at these schools, so this would be low poverty schools, so only 10% of students are eligible for free or reduced price plunge compared Thio teachers at a school where 90% of students are eligible for free lunch. Just the high high poverty school. Right? And I just did these calculations that teachers at schools at these, uh, this Assuming that the school has a 10% free lunch students, they can expect a 6% higher average salary compared to teachers at schools with 90% of students eligible for free lunch. That's a pretty big difference there. So again, that's just all. All that is to say that teachers air not not being compensated for teaching students from disadvantaged backgrounds. All right, we're nearly there. One more part left, Super quick. This is just the last part. Parts upside its part six, There you go. Last part is asking, is the pattern of results that you find here are consistent with the pattern in table 4.1? And what I find is no one big difference that I found is that the coefficient for the log of enrollment variable. It's negative in our results here, but is positive and table four point point and not entirely sure why that cases. But that is what it picked up so the log of enrollment coefficient is negative here, but it is positive in table 4.1. All right, Hopefully this will help you with this problem, and that's that's it for the six part problem.

Hi. There were working on number 14 Chapter one, section seven. Um, this one has four parts, but a lot of them are things that we just sort of have to perform. Tasks that we have to perform. Um, it gives us a chart with some numbers in it, and it's gonna ask us to do a few things. First. It's gonna ask us, Thio create a scatter plot of the data using a graphing utility. That means we don't have to draw necessarily. So I just went in, search up the first thing that Google had to offer and put in the information. Um, actually, it worked out pretty nicely. According to this, I've got t equals nine, corresponding to 1999 that each one of these are successive years Afterwards. Um, I used the regression feature, So that was part B. And I found that Latino model of doing the line for me and that it shows them both at the same time. It actually fits pretty well. Let me go over here to my whiteboard. I'll just go ahead and check off. Yes, I did a and yes, I did, B. Okay. And then part see it says plot the data in graph in the same viewing window, which we did. And the question just ask, is the model of good fit? Yes, it is. It also asks to explain it. Um, it pretty much lines up with all the dots. It is pretty much equidistant from all the dots closer to some and farther from others. But generally speaking, it's a good fit. That's part C. So we've got that ones. But check that one off a swell. Um hmm. Next, use the model and I should use that to predict some salaries for the future. Asks for 2005 and 2010. Um, I could do that. What I can do is I can take from this. If you look down here at the bottom, you can see that it's got the equation. That equation is the equation before the line that it drew all the way up here just very last row right here. So when they hadn't took that and type it into my Texas Instruments ti i 89 simulator and put it in Y one right here, just as I got it straight from the from that website And then I'm gonna go ahead and look at the table to see what the values are so you could do a number of different ways. I like doing like this because I can kind of scroll through it. First thing I want to look for is the value for 2005 which would correspond to t equals 15. And that would be 446564 Let's write that down. Okay. And the other one that asked for Waas, Um, a 2000 and 10 2010 would be it would be corresponding with 20. So 50.778 50.778 Now they're saying that these air in thousands of dollars So these air decimals right here. But if you could imagine that those were commas that would show you what the annual salary would be. Eso they say, Do the results seem reasonable? Well, yeah, they do. Well, I mean, obviously they should pay teachers quite a bit more, don't you think? But it looks to me like it's pretty much in line with the trend and s Oh, yeah. It looks exactly what exactly on par for for this exact model, So yeah, it seems very reasonable. My explanation is it fits with the with previous plot points. Thanks.

Or I saw the main salaries from 3 4007 are showing a table. A model that approximates the state out is given by. So this is the motto that approximate that data 42.6 minus 1.95 T well, minus point point 0.0 60. All right, so so. Question A is used to graph intimidated the critic scatter point of this data. All right, so let us use, uh, whatever eternity to plug this data. So after plotting the data is gonna be something like So the ex excess is gonna be zero two for six. Spread up evenly. And here is 40. Yeah, here, we got something like 60. All right, so this the poor, we got a little bit more than here. Something like ear here. Here for here. It's Ah, it's so is increasing. All right, so it's something like it's not that accurate, but it's approximately like this, right? So we have points, points, points here, points their points here, points there, and points here at Central. All right, so that's the grift from ah, graphic entity, whatever you want. And the question is how well does the mother fit the data? Or so the quick answer is the model fits the data. Ah, very well. Says all All of the points are close to the model, right? It's all of the points of coast to the model. So fits. Well, our question see, is that according to the motto, in what year with the salary for classroom teachers, exit 16 k, so 60 k. So we're gonna have a hard on ally from 60 K so that we can approximate the year here. So this is what we want. All right. From this graph, it's gonna be, um, roughly the tea. Miss Rafi here, we're gonna have this value. It's gonna be 10 point 545 right? So the well, I have to say that the gravity is terrible because this should have bean tin something. Should I be 10 something? But it's in the graph. It's gonna be for something. It's because that your autograph badly the two the two here should be do 446 and 10 here. So it's gonna be so this value remember, that is gonna be 10.545 Just ignored the graph e. It's terrible on the access ourself from the graph the Model X it's 60 k. It's gonna be at T approximately 11 years exactly 10.5 for five years. According Thio corresponds to to side than it 11. All right, so question D is that Is this model valid for long term predictions of the classroom Teacher salaries? Well, uh, whether the answer is the moderates now, valid price of the moderates not fell it for a long term prediction because it increases a lot more rapidly as time goes on that then it does in the very beginning of this model because right, so you can see that a slope of this graph increased slowly. So it's gonna be in the future is like this, right? So it's not gonna be feeding that the model. The model has a very go, as I'm told at, like, t ecause 17 or something which causes tomato to increase without bond as Ticos closer, closer to 16 um, which is no re elastic. So the Varda grass, until we also made that a 2016 a teacher salary would not exist. So by all of the reasons here, the motto is not valid for a non term predictions


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