MDM 4UI: Unit 8 Day 2: Regression and Correlation

Transcription

1 MDM 4UI: Unit 8 Day 2: Regression and Correlation Regression: The process of fitting a line or a curve to a set of data. Coefficient of Correlation(r): This is a value between and allows statisticians to identify whether a linear correlation exists and so, classify it as / and /. The classification scale is as follows: Negative Linear Correlation Positive Linear Correlation Perfect Strong Moderate Weak Weak Moderate Strong Perfect Coefficient of Determination(r 2 ): Gives the relative strength of the relationship (what percentage of the change in the dependent variable is due to changes in the independent variable) Denoted r 2 Is always a positive number from 0 to 1 which represents the percentage described above If r 2 = 0.44, then 44% of the changes in the dependent variable are due to changes in the independent variable. The closer 1, the stronger the relationship. The following diagram illustrates how the correlation coefficient corresponds to the strength of a linear correlation. USING DESMOS 1. Go to and Launch Calculator Click here and select Table

2 2. Enter the following data into the table of values. Make sure your independent value is x and your dependent value is y. Clicking on the wrench in the top right will allow you to choose appropriate ranges for the x- and y-variables. Price ( /L) Amount Sold (L/h) You need to tell the program to estimate a line of best fit. Choose to add an f(x) expression it will put your curser in a new equation box. Type: y1~mx1+b (the reference numbers will shrink when you type them). Your line of best fit will appear. Click here and choose f(x) expression Type y1~mx1+b (The numbers will shrink on their own) Your line of best fit should appear.

3 4. Now we need to interpret the information. This box gives you the equation of the curve. The values for m and b need to be inserted into the equation. We can now write the equation of the line of best fit. You should round decimals to two decimal places: y = x + We should then interpret this with respect to our data: = 13.08( ) There are also others value calculated: r 2 = r = These are the coefficient of determination (r 2 ) and the coefficient of correlation (r). The coefficient of determination tells us: The coefficient of correlation tells us: RESIDUALS The r-value (correlation coefficient) indicates the strength and direction of a linear relationship, however a lower r-value does not necessarily mean that the linear model (line of best fit) should be rejected. Another way to analyze how well the line fits the data is to calculate the vertical distance between each data point and the line of best fit. If the line is a good fit, these values should be fairly small and there should be no noticeable pattern. The vertical distance between a data point and the line of best fit is called the.

4 It can be calculated for a single point ( ) x, y 1 1 by subtracting the calculated y-value (using the line of best fit equation) from the actual y1value. EXAMPLE R 1 = y1 [ m( x1 ) + b] where m is the slope and b is the y-intercept of the line of best fit. Using the equation of the line of best fit from our Consumption vs Price data, calculate the residual value for (95.9, 172) data point. Line of Best Fit Equation: Litres = Price + m b The data point: (price, litres) = ( ) The Residual Value: R = y m( x ) + ] Residual Plots x, y 1 1 = (, ) 1 1 [ 1 b = = A residual plot is a graph that shows the residuals on the vertical axis and the independent variable on the horizontal axis. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a non-linear model is more appropriate. If the residual plot shows a fairly random pattern (ie. the first residual is positive, the next two are negative, the fourth is positive, and the last residual is negative) then the residuals indicate that a linear model provides a good fit to the data. Below, the residual plots show three typical patterns. The first plot shows a random pattern, indicating a good fit for a linear model. The other plot patterns are non-random (U-shaped and inverted U), suggesting a better fit for a non-linear model. Random pattern Non-random: U-shaped Non-random: Inverted U

5 USING DESMOS TO DETERMINE RESIDUALS Desmos allows you to calculate and graph the residual values for all your data. Click here to add the residual values to your table of values and to plot the residuals. These will plot along the x-axis. How good was our line of best fit? Explain.

6 Example: The following table shows the average wage in each of the ten regions in Canada, as well as the average number of working days lost per worker in each region due to any cause(e.g., illness, or injury). The data is adapted from Statistics Canada for 2007 Average Wage ($/h) Days Lost (days) Determine the linear coefficient of determination and the correlation coefficient to the nearest 100 th. Interpret what these values tell you about the relationship. Use the equation of the line to predict the average wage if 5 days are lost. Plot the residuals. What does this tell you about the line of best fit? Homework: p #3 5, 8, 9, 13