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.