Linear Regression

Objective Find the Line of Regression Use Line of Regression to Make Predictions

Relevance To be able to find an equation to best represent quantitative data with 2 variables and use it to make predictions.

Warm Up In many communities there is a strong positive correlation between the amount of ice cream sold in a given month and the number of drownings that occur in that month. Does this mean that ice cream causes drowning? If not, can you think of other alternatives for the strong association?

Warm Up #2…… Explain why one would expect to find a positive correlation between the number of fire engines that respond to a fire and the amount of damage done in the fire.

Regression Line…… If the value of the correlation coefficient is significant, the next step is to find the equation of the regression line. Regression Line – The data’s line of best fit which is determined by the slope and y-intercept.

Regression Analysis…… It finds the equation of the line that best describes the relationship between the 2 variables. Primary Purpose: To Make Predictions *This is a test question.

Prediction Models……

Remember Algebra?...... The slope intercept form of a line was y = mx + b where m is the slope and b is the y-intercept Slope: The change in y over the change in x. y-intercept: where the line crosses the y-axis.

Line of Best Fit…… The equation used to find the line of best fit is y = ax + b where “a” = slope and “b” = y-intercept

Computational Formulas……y = ax + b To find a: To find b:

Example 1…… Find the equation of the line of best fit. Predict the # of sales when 5 ads are sold. # of ads # of sales 3 7 4 6 2 5 10 8

Go by the formula……These are the lists you will need.

First…… Find the mean of x and the mean of y and write it down. Put x’s in L1 – stat calc one var stats L1 Put y’s in L2 – stat calc one var stats L2

Means of x and y……

Let’s fill in the lists…… L5 =L3 x L4 L6 = L3 squared x y x - xbar y - ybar (x-xbar)(y-ybar) (x - xbar) squared 3 7 -0.8 -0.2 0.16 0.64 4 6 0.2 -1.2 -0.24 0.04 2 5 -1.8 -2.2 3.96 3.24 10 2.2 2.8 6.16 4.84 8 0.8 10.2 8.8

Compute “a”……

Compute “b”……

Plug into y = ax + b…… Answer: y = 1.16x + 2.8

Predict …… Predict the number of sales when 5 ads are sold. Y = 1.16(5) + 2.8 = 8.6 = 9 sales

Example 2…… A. Find the equation of the line of best fit. B. Predict hours of exercise if the person is 35 yrs old. C. Predict the age if they exercise 9 hours per week. Age Exercise 18 10 26 5 32 2 38 3 52 1.5 59 1

Find the means…… X-Values: Y-Values:

The lists…… L1 L2 L3 = L1 - 37.5 L4 = L2 - 3.75 L5 =L3 x L4 L6 = L3 squared x y x - xbar y - ybar (x-xbar)(y-ybar) (x - xbar) squared 18 10 -19.5 6.25 -121.9 380.25 26 5 -11.5 1.25 -14.38 132.25 32 2 -5.5 -1.75 9.625 30.25 38 3 0.5 -0.75 -0.375 0.25 52 1.5 14.5 -2.25 -32.63 210.25 59 1 21.5 -2.75 -59.13 462.25 -218.75 1215.5

Compute “a” and “b”……

Equation: y = mx + b Plug into the formula for the equation of the trend line. Y = -.18x + 10.50

Predictions…… Find y when x = 35. Y = -.18(35) + 10.50 Y = 4.2 hours Find x when y = 9. 9 = -.18x + 10.50 9-10.50 = -.18x -1.5 = -.18x X = 8.3 X = 8 years