Estimating with trend lines
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Shira's math test included a survey question asking how many hours students had spent studying for the test. The graph below shows the relationship between how many hours students spent studying and their score on the test. Shira drew the line below to show the trend in the data. Assuming the line is correct, what does the line slope of 15 mean? So let's see. The horizontal axis is time studying in hours. The vertical axis is scores on the test. And each of these blue dots represent the time and the score for a given student. So this student right over here spent-- I don't know, it looks like they spend about 0.6 hours studying. And they didn't do too well on the exam. They look like they got below a 45, looks like a 43 or a 44 on the exam. This student over here spent almost 4 and 1/2 hours studying and got, looks like, a 94, close to a 95 on the exam. And what Shira did is try to draw a line that tries to fit this data. And it seems like it does a pretty good job of at least showing the trend in the data. Now, slope of 15 means that if I'm on the line-- so let's say I'm here-- and if I increase in the horizontal direction by 1-- so there, I increase the horizontal direction by 1-- I should be increasing in the vertical direction by 15. And you see that. If we increase by one hour here, we increase by 15% on the test. Now, what that means is that the trend it shows is that, in general, along this trend, if someone studies an extra hour, then if we're going with that trend, then, hey, it seems reasonable that they might expect to see a 15% gain on their test. Now, let's see which of these are consistent. In general, students who didn't study at all got scores of about 15 on the test. Well, let's see. This is neither true-- these are the people who didn't study at all, and they didn't get a 15 on the test. And that's definitely not what this 15 implies. This doesn't say what the people who didn't study at all get. So this one is not true. That one is not true. Let's try this one. If one student studied for one hour more than another student, the student who studied more got exactly 15 more points on the test. Well, this is getting closer to the spirit of what the slope means. But this word "exactly" is what, at least in my mind, messes this choice up. Because this isn't saying that it's a guarantee that if you study an hour extra that you'll get 15% more on the test. This is just saying that this is the general trend that this line is seeing. So it's not guaranteed. For example, we could find this student here who studied exactly two hours. And if we look at the students who studied for three hours, well, there's no one exactly at three hours. But some of them-- so this was, let's see, the student who was at two hours. You go to three hours, there's no one exactly there. But there's going to be students who got better than what would be expected and students who might get a little bit worse. Notice, there's points above the trend line, and there's points below the trend line. So this "exactly," you can't say it's guaranteed an hour more turns into 15%. Let's try this choice. In general, studying for one extra hour was associated with a 15-point improvement in test score. That feels about right. In general, studying for 15 extra hours was associated with a 1-point improvement in test score. Well, no, that would get the slope the other way around. So that's definitely not the case. So let's check our answer. And we got it right.