Linear+least+squares+regression,+plus+quadratic+and+exponential+regression

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If using Least Squares Regression, you will be using a scatterplot. You will then need to understand Correlation. Correlation is a measure of the strength of a linear relationship. There are two ways to find correlation (which is represented using an "r").
 * Linear Least Squares Regression Plus Quadratic and Exponential Regression** Regression Analysis is the study of the relationship between quantitative variables. The purpose of regression is to help us predict the values for different variables. There are three types of regression.======
 * 1) **Least Squares Linear Regression**- Used when a line is the best fit for the data.
 * 2) **Quadratic Regression-** Used when a parabola is the best fit for the data.
 * 3) **Exponential Regression**- Used when one must convert an expential equation to a linear one.
 * Least Squares Linear Regression**
 * Using Below Formula


 * Plugging it into your calculator. Below are steps on how to find "r" using your calculator.
 * 1) Plug in your two sets of numbers using the "Stat Plot" key. (For the sake of this example, we'll make the independent variable in L1 and the Explanatory Variable in L2.
 * 2) Once you have your two lists and the amount of numbers in both list are equal quit and go to your main menu.
 * 3) Punch in the following: "Stat", "Calc" (Scroll along the top from "Edit" to "Calc", "LinReg (ax+b)". Once you click on LinReg you will have the option to punch in your lists. Punch L1, and then L2, separated by a comma. Press enter and you will get a series of numbers. One of which will be r.
 * Some Notes On R**
 * R will be in between -1 and 1. A negative r implies a negatively sloped plot. A positive r implies the opposite. A horizontal line will have a slope of 0.
 * -1 and 1 represent strong correlations (this means that the line of best fit will be a good fit to the graph)
 * Correlations with the higest ABSOLUTE VALUE will have the strongest correlation. Therefore, -.8 and .8 are equal strength correlations.
 * R cannot be used for exponential relationships. Ever.
 * Squaring the r value and multiplying it by 100 will give you the percent of variation in the data.
 * R is a resistant value.
 * How to find the line of best fit**
 * The line of best fit is represented by the following equation: ŷ=ax +b
 * "a" and "b" are constants. Put in any "x" or "y" value to find out what the predicted variable will be.
 * "a" is found by the following: a=r * (Sy/Sx). This will give you the slope of the line
 * "b" is found by the following: b= y-average - (a* x-average)
 * Another easier way to find the line of best fit is to use the above steps used to find "r". When you plug in the numbers and press enter you will also get the equation for the line of best fit. (If before pressing enter you put another comma and then add Y1, the calculator will graph the equation for you.
 * The point (average value of x, average value of y) should always appear on the line of best fit. If it doesn't you did something wrong.
 * Is it a good fit?**
 * After finding the line of best fit you are able to tell if the line is a good fit or not.
 * If the value has a strong value of "r" (meaning it is close to 1 or -1) and the line follows along with the points in the scattergram it should seem as though it is a good fit. However, there are occasions where the graph and "r" are misleading.
 * Using a residual plot helps you tell if the line of best fit is actually a good fit to the line. To learn about residuals click [|here].
 * Sometimes, by using a residual, you can see that the points are following a pattern, and the line would not be a good fit with points that are not listed.
 * Quadratic Regression**
 * For quadratic regression, a line will no longer be the best fit for the data. Instead a parabola will be.
 * Quadratic regression is represented by the following equation: y= ax^2 + bx+ c
 * To find "r", "a", "b", and "c" plug your sets of data into your calculator. Use the steps used for linear regression except instead type in quadratic regression to get your values.
 * From there you can use are and compare the parabola to the data to see if the parabola is a good fit.
 * Exponential Regression**
 * The goal of exponential regression is to take a set of data that is best represented by the equation: y=ab^x
 * Use the same steps above to find "a" "b" and "r" except instead use "ExpReg" on your calculator.
 * The next step is to convert x and y values so that the data can be represented by a linear equation.
 * To do this instead of plotting the point (x,y), plot (x,log y). This will give you a linear equation.
 * The "a" of the new equation will be 10^slope
 * The new "b" will be 10 ^ y intercept.

1. Below is a table of the ages of wives vs. the ages of husbands at the time of marriage. Find the correlation coefficient. What is predicted age of a husband when the wife's age is 31. Part 1. Use your "LinReg" key on your calculator to find the value for r. r=.9439 Part 2. Use the same "Lin Reg" key to find the least squares regression line. y=1.013595523x + 1.415018127 Plug in 31 for x. y= 32.56 years for the husband.
 * Examples**
 * Wife || Husband ||
 * 22 || 25 ||
 * 32 || 25 ||
 * 50 || 51 ||
 * 25 || 25 ||
 * 33 || 38 ||
 * 27 || 30 ||
 * 45 || 60 ||
 * 47 || 54 ||
 * 30 || 31 ||
 * 44 || 54 ||
 * 23 || 23 ||
 * 39 || 34 ||
 * 24 || 25 ||
 * 22 || 23 ||
 * 16 || 19 ||
 * 73 || 71 ||
 * 27 || 26 ||
 * 36 || 31 ||
 * 24 || 26 ||
 * 60 || 62 ||
 * 26 || 29 ||
 * 23 || 31 ||
 * 28 || 29 ||
 * 36 || 35 ||


 * SAT/ AP Style Questions**

1. Data are obtained from a group of high school seniors comparing age and the number of hours spent on the telephone. The resulting equation is: y=.123x +2.57 with an r value of .866. What percentage of variation can be explained by this model? a. 0.75% b. 0.866% c. 75% d. 86.6% e. this value cannot be determined

2. What is the effect of the correlation between two variables (x,y) if each x-value is cut in half and 0.04 is subtracted from each y-value? a. The correlation is cut in half b. The correlation is decreased by .04 c. The correlation is decreased by .0016 d. the correlation is unchanged e. the correlation is doubled

3. In a statistics course, a linear regression equation was computed to predict the final exam score based on the score on the first test of the term. The equation was y=.7x + 25. George scored an 80 on the first test. On The final test George scored an 85. What is the value of his residual? a. -4 b. 4 c. 4.5 d. 5 e. 81

1- c 2- d 3- b
 * Answers**

[|**http://www.sixsigmafirst.com/regression.htm**] [|**http://www.socialresearchmethods.net/kb/statcorr.php**] [|**http://stattrek.com/AP-Statistics-1/Residual.aspx?Tutorial=Stat**] [|**http://www.zweigmedia.com/RealWorld/calctopic1/regression.html**] [|**http://calculator.maconstate.edu/quad_regression/index.html**] [|**http://mathbits.com/Mathbits/TISection/Statistics2/exponential.htm**] Mr. Postman's Notes [|**http://www.pascack.k12.nj.us/70271913213132/blank/browse.asp?A=383&BMDRN=2000&BCOB=0&C=52119**]
 * Works Cited**