A.3.4 Lesson Summary
While working in math class, it can be easy to forget that reality is somewhat messy. Not all oranges weigh exactly the same amount, beans have different lengths, and even the same person running a race multiple times will probably have different finishing times. We can approximate these messy situations with more precise mathematical tools to better understand what is happening. We can also predict or estimate additional results as long as we continue to keep in mind that reality will vary a little bit from what our mathematical model predicts.
For example, the data in this scatter plot represents the price of a package of broccoli and its weight. The data can be modeled by a line given by the equation y = 0.46x + 0.92. The data does not all fall on the line because there may be factors other than weight that go into the price, such as: the quality of the broccoli, the region where the package is sold, and any discounts happening in the store.
We can interpret the y-intercept of the line as the price for the package without any broccoli (which might include the cost of things like preparing the package and shipping costs for getting the vegetable to the store). In many situations, the data may not follow the same linear model farther away from the given data, especially as one variable gets close to zero. For this reason, the interpretation of the y-intercept should always be considered in context to determine if it is reasonable to make sense of the value in that way.
We can also interpret the slope as the approximate increase in price of the package for the addition of 1 pound of broccoli to the package.
The equation also allows us to predict additional values for the price of a package of broccoli for packages that have weights near the weights observed in the data set. For example, even though the data does not include the price of a package that contains 1.7 pounds of broccoli, we can predict the price to be about $1.70 based on the equation of the line, since 0.46*1.7 + 0.92 = 1.70.
On the other hand, it does not make sense to predict the price of 1,000 pounds of broccoli with this data, because there may be many more factors that will influence the pricing of packages that far away from the data presented here.
We can interpret the y-intercept of the line as the price for the package without any broccoli (which might include the cost of things like preparing the package and shipping costs for getting the vegetable to the store). In many situations, the data may not follow the same linear model farther away from the given data, especially as one variable gets close to zero. For this reason, the interpretation of the y-intercept should always be considered in context to determine if it is reasonable to make sense of the value in that way.
We can also interpret the slope as the approximate increase in price of the package for the addition of 1 pound of broccoli to the package.
The equation also allows us to predict additional values for the price of a package of broccoli for packages that have weights near the weights observed in the data set. For example, even though the data does not include the price of a package that contains 1.7 pounds of broccoli, we can predict the price to be about $1.70 based on the equation of the line, since 0.46*1.7 + 0.92 = 1.70.
On the other hand, it does not make sense to predict the price of 1,000 pounds of broccoli with this data, because there may be many more factors that will influence the pricing of packages that far away from the data presented here.