Linear RegressionLinear RegressionLinear RegressionLinear Regression

Section 3.3Section 3.3

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……

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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:

2)(

))((

xx

yyxxa

)( xayb

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

6 10

4 8

Go by the formula……These are the lists you

will need.

x y xx yy ))(( yyxx 2)( xx

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……L1 L2 L3 = L1 - 3.8 L4 = L2 - 7.2 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

6 10 2.2 2.8 6.16 4.84

4 8 0.2 0.8 0.16 0.04

10.2 8.8

Compute “a”……

16.1159090909.18.8

2.10a

Compute “b”……

8.2)8.3)(159090909.1(2.7 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”……

18.5.1215

75.218

a 50.10)5.37)(18.(75.3 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