Interpreting a trend line | Data and modeling | 8th grade | Khan Academy


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.

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