Line of Best Fit

Warm-Up Write the equation of the line passing through each pair of passing points in slope- intercept form. 1. (5, –1), (0, –3) 2. (8, 5), (–8, 7) Use the equation y = –0.2x + 4. Find x for each given value of y. 3. y = 7 4. y = 3.5

Types of Correlation Strong Positive Correlation – the values go up from left to right and are linear. Weak Positive Correlation - the values go up from left to right and appear to be linear. Strong Negative Correlation – the values go down from left to right and are linear. Weak Negative Correlation - the values go down from left to right and appear to be linear. No Correlation – no evidence of a line at all.

Positive Correlation

Negative Correlation

No Correlation

What is the correlation? The scatter plot shows a positive correlation between hours of studying and test scores. This means that as the hours of studying increased, the test scores tended to increase.

The scatter plot shows a negative correlation between hours of television watched and test scores. that as the hours of television This means that as the hours of television watched increased, the test scores tended to decrease.

What is a Line of Best Fit? When data is displayed with a scatter plot, it is often useful to attempt to represent that data with the equation of a straight line for purposes of predicting values that may not be displayed on the plot. Such a straight line is called the "line of best fit." It may also be called a "trend" line. A line of best fit is a straight line that best represents the data on a scatter plot.  This line may pass through some of the points, none of the points, or all of the points.

Line of Best Fit & Equation Step 1: Plot the data points. Step 2: Identify the correlation. Step 3: Draw the line of best fit, showing the general trend of the line. Step 3: Choose two points on the line of best fit, the points may not necessarily be a data point. Step 4: Find the slope using those two points Step 5: Use the slope and one of the points to substitute into y = mx + b and solve for b. Step 7: Write the equation of the line in slope-intercept form by substituting m and b into y = mx + b Try to have about the same number of points above and below the line of best fit. Helpful Hint

Example 1: Meteorology Application Albany and Sydney are about the same distance from the equator. Make a scatter plot with Albany’s temperature as the independent variable (the “x” coordinate). Name the type of correlation. Then sketch a line of best fit and find its equation.

• • • • • • • • • • • • Example 1 Continued Step 1 Plot the data points. Notice that the data set is negatively correlated...as temperature rises in Albany, it falls in Sydney. Step 2 Identify the correlation. Step 3 Sketch a Line of Best Fit o • • • • • • • • • • • •

Step 4 Identify two points on the line. Example 1 Continued Step 4 Identify two points on the line. For this data, you might select (60,52) and (40,61). Step 5 Find the slope of the line that models the data. 61 – 52 40 - 60 9 -20 -.45 Step 6 Use the slope and one of the points to substitute into y=mx+b. y= mx + b 52 = -.45(60) + b y = –0.45x + 79 52 = -27 + b 52 + 27 = b 79 = b Step 7 Write the equation of the line in slope-intercept form: y = –0.45x + 79

Example 2 Make a scatter plot for this set of data. Identify the correlation, sketch a line of best fit, and find its equation.

• • • • • • • • • • Example 2 Continued Step 1 Plot the data points. Notice that the data set is positively correlated…as time increases, more points are scored Step 2 Identify the correlation. Step 3 Sketch a Line of Best Fit • • • • • • • • • •

Example 2 Continued Step 4 Identify two points on the line. For this data, you might select (20, 10) and (40, 25). Step 5 Find the slope of the line that models the data. Step 6 Develop the equation: y= m(x) + b 10= .75(20) + b 10= 15 + b -5= b Step 7 Write the equation of the line: y = 0.75x - 5

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Practice with Round Robin Each Student has a colored pencil and a piece of graph paper. Each student does step 1 (draw the scatter plot-Round to the nearest hundred) of a different problem UCLA cost trend 1) Living on campus in state 2) Living on campus out of state 3) Living off campus in state 4) Living off campus out of state . Pass your paper in a clockwise direction – coaches the person if the work received was incorrect, praise when correct. Each student now completes step 3 (draw a line of best fit) and initials their work. Each student now completes step 4 & 5 (identify two points and find the slope) and initials their work. Each student now completes step 6-7 and initials their work.