Often times, neither a line nor quadratic function is a good fit for the data, and so a higher order polynomial function is needed (as in the tiger population example illustrated in the overview of this module). This is one of the main applications of polynomials, to use them to find a curve that best fits the data, in what is called the "least squares" sense. This line of best fit is also called a regression line.
The model obtained can be used not only to make predictions but also to illustrate behavior between the independent and dependent variables. While the idea is essentially the same no matter what order of polynomial is used, in this activity, we will focus on the simplest case, the linear function, y = mx + b, which is a 1st -degree polynomial.
Please read through this discussion and think about these questions as you proceed through the next few sections. When you have a good understanding of the concepts, begin your search for the data needed to answer the questions below. When you are confident with your research and outcomes, return to this discussion and post a thorough response to one of the following questions.
Now, let's begin our graphing discussion:
1. Consult an almanac, newspaper, magazine, or the Internet to find data that appear to lie approximately on or near a straight line. A few of the many free sources of data that might be appropriate to use are attached:
2. Create a scatterplot and a line fitting the data in one of two ways:
a. To do this activity by hand, create a scatterplot by plotting the (x, y) data in a Cartesian (rectangular) coordinate system. Then, draw a line that approximately fits the data in the scatterplot, and then write its equation as a function in slope-intercept form. An easier way, and one that produces better looking results, is to find the equation of the line for your data using this free, online linear regression calculator:
b. To use the regression calculator:
Enter your data, but without parentheses ( ), in the empty data field
When all of your data points are entered, click on the "Submit Data" button. The line of best fit y = mx + b for your data will be displayed, as will a figure showing a scatterplot of your data and the graph of your line of best fit.
Right-click on the displayed figure of the scatterplot/graph and save it to your computer. Then, load it into the discussion box when you make your post. To do that, click on the image button in the palette of the discussion text-editor window. In that menu, you load the figure from your computer into your post.
3. Also include a link to your data, along with an explanation of what the x and y axes represent in your graph.
Part 2 if this were present
1.) Consider the data and regression line if present and try to explain some of the possible reasons for the observed variation, that is, what are some of the possible reasons the points do not all fall on the line? For example, if we were to plot the number of births of a certain species of bird each year (dependent variable) vs. the annual rainfall in inches (independent variable) in their habitat that year, we will find a linear, though not perfectly so, relationship. The reason for the variation (vertical offset from the line) may be in part because, for lower than average rainfall there could be a food shortage.
Attachment:- wk.rar