Comparing models to fit data example
- Christine works in a movie store in her hometown. Using the store's total selection, she documented the price of each movie title and how many years it has been since it was featured in movie theatres. She plotted the points below. So let's see what's going on below here. Looks like there's two curves that she tries to fit. I'm assuming we're going to read about it in a second. But these blue points are the data points. So, for example, this data point right over here shows a movie that the title costs six dollars, and it has been released for almost two years, a little under two years. This data point right over here, this is a movie that has been released for almost four years, looks like maybe three and three quarters years. And they're selling that, looks like for a dollar or even a little bit less than a dollar. So those are her data points. So once, again, she documented the price of each movie title as a function of how many years it's been since it was featured in movie theatres. She is looking for a function that models her data. Since the trend of the data is decreasing and convex, and you see it here, it's definitely decreasing, and convex, it's opening upwards, if you imagine a curve, it looks like it's opening upwards a little bit like that, so decreasing and convex, she found a decreasing convex exponential model and a decreasing convex quadratic model. Which of the following functions better fits the data? Function A, this is an exponential. This is the one in green right over here. And Function B, this one right over here is a quadratic. And you can see this one in purple. And so, which one of those better fits the data? If we look at what's going on here, the green function, the exponential one, most of the data points for any given duration, for how long the title's been out, it looks like it's consistently underestimating. That it's always, the model's guess, or what the model would say the price is, is always, essentially except for only one data point right over here, for all of these other data points it's underestimating what the price would be. The purple model or the purple function right over here, it has more of a balance between overestimating, right over here, it's overestimating by a little bit, and underestimating. And its underestimates are closer, and its overestimates are closer than this green model. So I would say that Function B is definitely a better model. Use the function of best fit, so we're going to say Function B, to predict the price of a movie that was featured in theatres 5.5 years ago. Round your answer to the nearest cent. So 5.5 years ago, that's going to be right over here. We're going to go to Function B, which is this purple one. So it's going to be under a dollar. But we want to get something to the nearest cent, so let's actually use the actual definition of the function. So this is price as a function of how long the movie has been released. Where x is how long it's been released, and y is its price. If x is 5.5, let's figure out what y is going to be. So y is going to be equal to 0.5 times x squared. So x is 5.5 squared. So then we have minus five times x again. So minus five times 5.5. And then we have plus 13. And what does that get us? That gets us 62 1/2 cents. If we were to round our answer to the nearest cent, that's going to be 63 cents. And we got it right.