Learning Goals. 2. To be able to distinguish between a dependent and independent variable.

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1 Learning Goals 1. To understand what a linear regression is. 2. To be able to distinguish between a dependent and independent variable. 3. To understand what the correlation coefficient measures. 4. To be able to describe the correlation coefficient accurately. 5. To understand what the coefficient of determination measures. 6. To be able to describe the coefficient of determination accurately. 7. To know how to perform a linear regression on the TI - 83 calculator. 8. To be able to explain what the equation of the line means in the context of the question. 8.1 Line of Best Fit Scatter Plot a graph of plotted points that show the relationship between two sets of data. The trend (a pattern of average behavior that occurs over time) in a scatter plot can often reveal the nature of the relationship between two variables as either negative or positive.

2 In data analysis you are often trying to determine whether one variable, the dependent variable, is affected by another, the independent variable. Independent Variable Always found along the x-axis of a graph. It is the variable that you choose the values for. Dependent Variable Always found along the y-axis. Its value is dependent on the value chosen for the independent variable. Linear Regression the formal (computerized) process by which the line of best fit is mathematically determined. Characteristics of a Line of Best Fit 1. A straight line that represents a trend in the scatter plot as long as the pattern is more or less linear. 2. Should pass through as many points as possible, with about half the points above and half below the line.

3 Linear Correlation a relationship in which a change in one variable tends to correspond to a proportional change in another variable. Correlation Coefficient (r) a measure of how well a linear model fits a two-variable set of data. Note: r can only be calculated for linear models. Correlation Coefficient (r) Possibilities 1. Values of r between -1 and 0 indicate a negative correlation, so the line of best fit has a negative slope. 2. An r value of 0 indicates that there is no linear correlation. 3. Values of r between 0 and 1 indicate a positive correlation, so the line of best fit has a positive slope.

4 It is important to be aware that increasing the number of data points used in determining a correlation improves the accuracy of the mathematical model. Example One Discuss the correlation for each relationship in terms of sign and strength.

5 Coefficient of Determination (r 2 ) A number from 0 to +1 that gives the relative strength of the relationship between two variables. For example, if r 2 = 0.44, this means that 44% of time, any changes in the dependent variable is because of a change in the independent variable. Note: r 2 can be calculated for linear and non-linear models.

6 Example Two Use the r values given in Example One to identify the r 2 value for each scatter plot. Describe what the r 2 value means to each graph. November 26, 2015 You can use the line of best fit to make predictions by interpolation (estimating between data points) and extrapolation (estimating beyond the range of the data).

7 Using Technology to Calculate the Line of Best Fit In class you will be responsible to know how to calculate the line of best fit, the correlation coefficient and the coefficient of determination using both the TI 83 calculator as well as MS Excel. Review how to calculate regression on a calculator handout. Example Three The table shows distance-time for a student who is walking in front of a motion sensor. d represents the distance between the walker and the motion sensor, in metres, after t seconds have passed. a) Create a scatter plot relating distance, d, and time, t, on the TI 83 calculators. b) Determine the strength of the linear correlation between these variables.

8 c) Determine the correlation coefficient (r 2 ) for this data. What does it mean in the context of this question? d) Determine the equation of the line of best fit and explain what it means in the context of this question. e) Use the linear model to predict (interpolate) how far the student is away from the motion sensor at t = 2.5. f) Use the linear model to predict (extrapolate) how far the student is away from the motion sensor at t = 8.

9 Complete: p. 390 #1 6. Complete p. 390 #5, 6 both on the calculator and in Excel.