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The data in MEAPOl are for the state of Michigan in the year $2001 .$ Use these data to answer the following questions.(i) Find the largest and smallest values of math. Does the range make sense? Explain.(ii) How many schools have a perfect pass rate on the math test? What percentage is the totalsample?(iii) How many schools have math pass rates of exactly 50$\% ?$(iv) Compare the average pass rates for the math and reading scores. Which test is harder to pass?(v) Find the correlation between $m a t h 4$ and read $4 .$ What do you conclude?(vi) The variable expenditure per pupil. Find the average of exppp along with its standarddeviation. Would you say there is wide variation in per pupil spending?(vii) Suppose School A spends $\$ 6,000$ per student and School $B$ spends $\$ 5,500$ per student. By what percentage does School A's spending exceed School $\mathrm{B}$ 's? Compare this to 100$\cdot[\log (6,000)-$$\log (5,500) ],$ which is the approximation percentage difference based on the difference in thenatural logs. (See Section A.4 in Appendix A.)

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Willhelm S.

March 2, 2021

I just checked and saw there are 1823 obs. for bcode and frequency is always 1, so never mind. Missunderstood the data set.

Dear Alex, your answer is right but I believe only because there are no schools with > 1 student who reached 100%. The question asks how many schools have a perfect pass rate, not how many students. So if there would have been more than one student per

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moving on to computer exercise number three. Ah, the first thing we need to do is import the data set M e a p 01 which had includes data for the state of Michigan in the year 2001 course with downloaded the data said from the official website that imported right here. Now, as we've explained before, the first thing we need to do is hit the command described which I should write briefly and see what kind of data said we're dealing with and what kind of observation of variables we have. All right, so we can see that we have 1823 observations with 11 variables and some of the main variables we're gonna use with this exercise is ah, math for and read for this, uh, this corresponds to the percentage of students I have a satisfactory performance and fourth grade math and reading, respectively. Another variable that we're gonna use his ex BP, which is the expenditure per pupil. And, uh, also we have the natural longer than it's ah mahn atomic transformation of this of this variable here. And as we can see, we also have this tricks codes and building coast. Where schools in Michigan. All right, The first question asked. Find the largest and smallest values of math. Four. Does that range? Make sense? Explain. Alright. Same old command. Summarize math four with details one. Learn as much as possible about the variables. All right, so here, in order to find the larges and smallest values and the range, which is the difference of the two, we're gonna look at the smallest here in the first person tell which is zero and the largest is 100. The meaning is around 71.9 72% off seasons have a satisfactory performance in fourth grade maths. So the range is 100 and zeros other rages. 100 is the highest range possible. Does this make sense? Well, each observation corresponds to a school in Michigan, right? So there is at least one school. In fact, there is only one school. You're going to find out its account if math for equal zero. So they're just one school that has a 0% of students with satisfactory performance in math. Four. Sad but true. So there is one school with a 0% of form straight, and the highest is 100%. Everyone is doing great in fourth grade math, and in fact, there enough schools with very good performance going to count it. My four. It was 100 38 schools. Not that. All right, so the range does make sense. Ah, it is a little bit off putting if you see that first. But it does make sense. All right. Question to how many schools have a perfect pass straight on the math test? What percentage? This of the total sample. Well, I just did that. This is the answer here. The perfect score would be 100% of students have a satisfactory performance. A perfect pass straight. Let's hit it again. This is 38 schools and what is the percentage of the total sample, while the totem sample we have it right here is 1823 observations. So we're gonna turn our stated a calculator with a display command display. What? 38 divided by the total assemble. 1821823 times 100 Have our answer. It's just 2.9%. Yeah, that's right. 2.2 point 1% of the sample that say has a perfect pass straight in Part C. How many schools have math past rates of exactly 50%? 50% is what is the medium is a fugitive percentile. I know it's not the 50 personnel. The distribution, but is, ah is the 50th percentile, the past rate. So to find out how many schools going to the same thing count if Month four is exactly equal to 15. So there's 17 schools, and even though the question doesn't ask how many school they are, we can give you the answer right here. 17. Divided by the total, it's less than 1% 0.93% of schools Okay heart for compare the average past rates for the math and reading scores. Which test is harder to pass? Of the first thing to compare, it's a samurai's the two variables here. We don't really need detail because we just want to look at the average, so to summarize math and summarize reads for okay, the average, uh, the average pass straight for the master has its 71.9. Let's say 72% of people past, but for the reading. It's 60 lesson, 61%. So what does this mean? This means that at least in 2001 the reading test was harder to pass because on average and remember, we're talking about on average. Ah, whenever is less People have this satisfactory performance in reading compared to maths, right around 10% indifference around 10%. Part five says, find the correlation between math foreign reads for what do you conclude here? Think them the thing it's implied, but by correlations, I mean the leaner correlation coefficient. Pearson court listen for efficient. Well, you don't buy a Greek letter row, and I'm pretty sure that's what it means. But should they should have mentioned explicitly all right and stay that the commander's core and followed by the two variables off course doesn't make any difference. If we put read four first math for first brother, Correlation is in variant to the order. Okay, here we get the correlation um, Matrix. Of course, we don't care about the diagonal elements because there's a correlation of the Braille with itself is always one, but we care about this number here, which reads 0.8 before 27. This is a very high number. This means that that of course, we remember that the correlation coefficient is between minus one and one, and the value of 0.8 before 27 means that the two variables very highly correlated in a positive way. What does this mean? It means that there's a very high degree of linear association. I want to remember that. But this kind of correlation play linear association. This is the underlying assumption. So, not surprisingly, schools. This means that schools that have high bass strait on one test of a strong tendency to have high pass traits from the other test. All right, this is what the association means. We do not impose. Which test goes first. We make no causal interpretation of that. It just means that people who tend to do good well on one test it didn't do well on the other or the school's actually, and this is not surprising at all, because those are two variables might be correlated with each other. But in fact they're correlated with another indulgence variable. Probably, you know, the quality of school, the, um, the overall performance of the students the number of hours that putting Thio studying, maybe even the i Q level, So it's no surprise at all. Okay, next question. Ah, the variable x p p is expenditure per pupil. Find the Evers X p p along with standard deviation. Would you say there is a wide variation in per pupil spending? Okay, here. We need to do summarize with detail, see what's going on. Excuse me. All right. So the your pupil is, um Ah, I guess, is in dollars. So it's 5194 point a list. 5000 $195. Understand? Aviation is 1000 and $92. Um, all right, so is this large sand aviation? Let's think about it. First of all, there's compute a 1% off. The mean San Division is so ah would display. What is it? I was in 92 divided by 12. Far. Sorry. I want 95 times 100 21%. Well, I would say that's not a huge but pretty substantial standard deviation on. Let's remember what the standard deviation, Um, measures. It's ah, measure. It's one of the standard measures of Central This look of dispersion around the mean and it's most ah, usual interpretation is the average distance of old data points around the mean. Now if we say that the average distance is like 1/5 of the mean, well, that's pretty substantial, right? Is not crazy is an incredibly This first butter data set is is something quite dispersed around the meat? Okay, enough. Our final question. Suppose school a spent $6000 per student in school be spent $5500 per student. But what percentage? The school a spending exceeds school be spending, compared this to the luck difference, which is the approximation percentage difference based on the difference in the natural logs. Okay, well, that's a very easy one. We just need to, uh, make a very simple computation. 6000 minus 200 100 divided by, uh by the 100 right times 100. So it's 9.1. Let's say 9.1% school A spends 9.8 point 1% more per student than school B. Now, let's compare this with the natural log here in st a to ah, you. The natural log would put Ellen and then the apprentices of 6000 minus measured love drink right second princes. It gives us 8.7%. Well, that's not too different. But it is different, right? The approximation. It's good, we're not too good. And this is because the natural log is a very good approximation with very small changes when very small changes occur. Uh, typically, I would say from my experience that it is pretty good around when the differences around 1% or less, and to illustrate what I mean. Here you see, we have that discrepancy of, Let's say, displaying nine 0.1, divided by 8.7 would have, ah, this pregnancy of 4.6%. But if we make the different small, So if we want to compute the let's say that said 5500 cycles school, he was 5000 950 right? So the difference is small, but we're talking about an actual difference of 0.84% and if we do the same with the natural log, we'll see that well, this is a pretty good approximation, almost identical. It's very, very small difference, and this is where best to use an actual. But for small changes

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