9.+Linear,+quadratic+and+exponential+regression

Regression
A regression is to find a best fit line to describe the form of the relationship. The name of the line is called the least squares regression line and it shows the relationship. The line is given in the form of an equation: y = mx + b for a straight line regression y = ax 2 + bx + c for a quadratic regression y = ab x + c for an exponential regression

Usually, the regression line will be created using a set of data. Subsequently, the equation of the line found through the regression process will be used to make a prediction of an output (y) value for a proposed value of the input (x).

Measures the strength of the linear relationship. It doesn’t show a curved relationship even if it’s a strong one. As R is closer to 0 a straight line is a poorer description of the data [bad fit], but when its close to either -1 or 1, it’s a strong fit. Below are some examples of correlation coefficients. The one all the way in the lower right corner when r = -.99 has the best fit of all 6 graphs. With such a high r, it shows how well the points actually fit to the line. The one in the upper left corner on the other hand has the worst fit since r = 0, it demonstrates no fit. R^2 is known as the coefficient of determination which is the proportion of the y values explained by the least squares regression line. A high R^2 is a good linear fit.
 * R (Correlation Coefficient )**

*Linear
The equation is, the slope = R * standard deviation of y / standard deviation of x, the means of x and y are points on the least squares regression line.

From the regression line, you can calculate the residuals. A residual is the predicted value from the regression line. A residual plot is a SCATTER PLOT of all the residuals.

**How to find Regression line**
1. Use a calculator and type your 2 lists of equal length. 2. Go to the CALC section under STATS and choose 4: LinReg (ax+b), press enter 3. Type your 2 lists, separated by a comma and then VARS, Y-VARS, Function and then Y1 4. your R and R^2 will appear and your least squares regression is what y=

- R - Residual plot If the residual plot shows no systematic pattern, it is good.
 * To determine if the line is good or not**



***Quadratic**
if the graph is quadratic/ exponential, have to change it to make it have a good linear fit

power functions, when x is raised to a power, y= 4x^3 the points that have a strong linear fit for a power function would be (log(x), log(y)); these are the points for a strong linear fit.

***Exponential**
Exponential function, y = 3^x A strong linear fit for exponential = (x, log (y))

1. Use a calculator and type your 2 lists of equal length. 2. Go to the CALC section under STATS and choose 0: ExpReg, press enter -Use this function on the calculator because if you were to use LinReg, then the line would not be a good representation of the data. 3. Type your 2 lists, separated by a comma and then VARS, Y-VARS, Function and then Y1 4. Reexpress the equation to make it linear using logarithms
 * How to Find the Equation Using the Calculator**

Example1 :

 * wife || husband ||
 * 22 || 25 ||
 * 32 || 25 ||
 * 50 || 51 ||
 * 25 || 25 ||
 * 33 || 38 ||
 * 27 || 30 ||
 * 45 || 60 ||
 * 47 || 54 ||
 * 30 || 31 ||

1. Find the equation of the Least Squares Regression Line, correlation coefficient, and coefficient of determination. 2. Using the Least Squares Regression Line, what is the predicted age of the husband whose wife is 50? What is the value of the residual?

Example 2:
1. What is the equation of the exponential graph? 2. Reexpress the equation as linear fit using logarithms. 3. Use the equation to predict the difference in temperature after 45 minutes.
 * Time || Difference in temp ||
 * 10 || 68 ||
 * 20 || 36 ||
 * 30 || 20 ||
 * 40 || 10 ||
 * 50 || 6 ||
 * 60 || 4 ||

Answers: Example 1 1. LSRL: y = 1.244X - 5.317 (equation that best matches the data) R = .921 (correlation coeff), R^2 = .849 2. y = (1.244)(50) - 5.317 = 56.883 years old Residual = O - P Residual = 51 - 56.883 Residual = -5.883

Example 2 1. LSRL: y = (114.055)(.944) X 2. ln(y) = -0.0576X + 4.737 3. ln(y) = (-0.0576)(45) + 4.737 ln(y) = 2.145 y = e^2.145 y = 8.542


 * Harder Questions**

A researcher uses a regression equation to predict home heating bills (dollar cost), based on home size (square feet). The correlation between predicted bills and home size is 0.70. What is the correct interpretation of this finding? (A) 70% of the variability in home heating bills can be explained by home size. (B) 49% of the variability in home heating bills can be explained by home size. (C) For each added square foot of home size, heating bills increased by 70 cents. (D) For each added square foot of home size, heating bills increased by 49 cents.  Answer: b- r = .7 therefore r^2 = .49 r^2 explains the proportion of values the least squares regression line represents. [|http://www.devexpress.com/Help/?document=XtraCharts/CustomDocument6231.htm&levelup=true] [|http://en.wikipedia.org/wiki/Linear_regression#Applications_of_linear_regressionhttp://www.stat.tamu.edu/~pkohli/303s mmer/ch10.pdfhttp://www.stat.tamu.edu/~pkohli/303summer/ch10.pdfhttp://stattrek.com/AP-Statistics-1/Regression.aspx?Tutorial=Stat]