Estimating with linear regression (linear models)

Liz's math test included a survey question asking how many hours students spent studying for the test the scatterplot below shows the relationship between how many our students been studying and their score on the test a line was fit to the data to model the relationship they don't tell us how the line was fit but this actually looks like a pretty good fit if I just eyeball it which of these linear equations best describes the given model so as you know this point right over here this shows that some students at least self-reported they studied a little bit more than half an hour and they didn't actually do that well on the test looks like they scored a 43 or 44 on the test this right over here shows our like this one over here is a student who says they studied two hours and looks like they scored about a 64 65 on the test and this over here or this over here took a study students who studied over four hours or they reported that and they got looks like a 95 or 96 on the exam and so then and these are all the different students each of these points is represents a student and then they fit a line and when they say which of these linear equations best describes the given model they're really saying which of these linear equations describes or is or is being plotted right over here by this line that's trying to fit to them that's trying to fit to the data so essentially we just want figure out what is the equation of this line well it looks like the y-intercept right over here is 20 and looks like all of these choices here have a y-intercept of 20 so that doesn't help us much but let's think about what the slope is when we increase by one when we increase along our x-axis by one so change in X is 1 what is our change in Y our change in Y looks like let's see we went from 20 to 40 it looks like we got went up by 20 so our change in Y over change in X for this model for this line that's trying to fit to the data is 20 over 1 so this is going to be our slope and if we look at all of these choices only this one has a slope of 20 so it would be this choice right over here based on this equation estimate the score for a student that spent three point eight hours studying so we would go to three point eight which is right over Alice II this would be three point eight would be right around here so let's estimate that score so if I go straight up where do we intersect our model where do we intersect our line so it looks like we would get a pretty high score let's see if I were to take it to the vertical axis it looks like they would get about a ninety seven so I would write my estimate is that they would get a ninety seven based on this model and once again this is only a model it's not a guarantee that if someone studies 3.8 hours they're going to get a ninety seven but it could give an indication of what maybe might be a reasonable to expect assuming be that the time studying is is the variable that matters but you also have to be careful with these models because it might imply if you kept going that if you get to study for if you study for nine hours you're gonna get a 200 on the exam even though something like that is impossible so you always have to be careful extrapolating with models and keep it ticket with a grain of salt this is just a model that's trying to fit to this data and you might be able to use it to estimate things or to maybe set some form of an expectation but take it all with a grain of salt