linear regression
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Linear Regression. Section 3.3. Warm Up. - PowerPoint PPT PresentationTRANSCRIPT
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Linear RegressionLinear RegressionLinear RegressionLinear Regression
Section 3.3Section 3.3
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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?
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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.
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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.
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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.
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Prediction Models……
baxyLineara :.
cbxaxyQuadraticb 2:.
)(:. xbaylExponentiac
xaycLogarithmid blog:.
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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.
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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
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Computational Formulas……y = ax +
b• To find a: • To find b:
2)(
))((
xx
yyxxa
)( xayb
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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
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Go by the formula……These are the lists you
will need.
x y xx yy ))(( yyxx 2)( xx
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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
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Means of x and y……
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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
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Compute “a”……
16.1159090909.18.8
2.10a
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Compute “b”……
8.2)8.3)(159090909.1(2.7 b
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Plug into y = ax + b……
• Answer:
y = 1.16x + 2.8
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Predict ……• Predict the number of sales when
5 ads are sold.
Y = 1.16(5) + 2.8 = 8.6 = 9 sales
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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
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Find the means……• X-Values: • Y-Values:
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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
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Compute “a” and “b”……
18.5.1215
75.218
a 50.10)5.37)(18.(75.3 b
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Equation: y = mx + b• Plug into the formula for the
equation of the trend line.
Y = -.18x + 10.50
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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