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2-2LINEAR REGRESSION
Be able to fit a regression line to a scatterplot.
Find and interpret correlation coefficients.Make predictions based on lines of best fit.
OBJECTIVES
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line of best fit linear regression line least squares line domain range interpolation
extrapolation correlation coefficient strong correlation weak correlation moderate correlation
Key Terms
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How can the past predict the future?
The trends shown by scatterplots can be used to predict the future. But making a prediction without a line of best fit to guide you would be arbitrary.
What can you tell about the sign of the correlation coefficient and the slope of the regression line?
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How can the past predict the future?
The domain is the set of 1st elements of an ordered pair.
The range is the set of the 2nd elements.If you use a number within the domain to predict
the y-value, that is called interpolation.If you use a number outside the domain to
predict the y-value, that is called extrapolation.
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Example 1 Find the equation of the linear regression line for
Rachael’s scatterplot in Example 1 from Lesson 2-1. Round the slope and y-intercept to the nearest hundredth. The points are given below.
(65, 102), (71, 133), (79, 144), (80, 161), (86, 191),(86, 207), (91, 235), (95, 237), (100, 243)
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How can the past predict the future?
The line of best fit is also called the linear regression line.
The line can be represented by an equation in the form of y = mx + b, where m is the slope and b is the y-intercept.
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Example 1 (cont.)
The linear regression equation is y = 4.44x - 187.67
60 65 70 75 80 85 90 95 100 10580
100120140160180200220240260
f(x) = 4.43824701195219 x − 187.666666666667
Number of Water Bottles Sold
Temperature (°F)
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Find the equation of the linear regression line of the scatterplot defined by these points: (1, 56), (2, 45), (4, 20), (3, 30), and (5, 9). Round the slope and y-intercept to the nearest hundredth.
CHECK YOUR UNDERSTANDING
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Example 2 Interpret the slope as a rate for Rachael’s linear
regression line. Use the equation y = 4.44x - 187.67
What is the slope?4.44
Slope is
Since y shows water bottle sales and x shows the temperature, the slope is a rate of bottles per degree.
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Example 2 (cont.)y = 4.44x - 187.67
The slope of 4.44 shows how many bottles she will sell for each degree the temperature rises.
Since she can’t sell a fraction of a bottle, she will sell approximately 4 bottles per degree.
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Using the equation y = 4.44x - 187.67,approximately how many more water bottles will Rachael sell if the temperature increases 2 degrees?
CHECK YOUR UNDERSTANDING
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EXAMPLE 3 Rachael is stocking her concession stand for a day in
which the temperature is expected to reach 106 degrees Fahrenheit. How many water bottles should she pack?
Substitute 106 for x in y = 4.44x - 187.67y = 4.44(106) - 187.67y = 282.97
Rachel should expect to sell about 283 bottles.This is extrapolation since 106 is not between 65 and
100, the low and high x-values.
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How many water bottles should Rachael pack if the temperature forecasted were 83 degrees? Is this an example of interpolation or extrapolation? Round to the nearest integer.
CHECK YOUR UNDERSTANDING
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How well the line is predicting the trend?
If most of the points are close to the line, it is a good predictor of the trend.
The correlation coefficient, r, is a number that is between -1 and 1, inclusive.
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How well the line is predicting the trend?
If the absolute value is greater than 0.75, there is a strong correlation.
If the absolute value is less than 0.3, there is a weak correlation.
Any other absolute value is a moderate correlation.
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How does the equation show the type of correlation?
Positive correlation coefficients show a positive correlation.
Negative correlation coefficients show a negative correlation.
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EXAMPLE 4 Find the correlation coefficient to the nearest hundredth
for the linear regression for Rachael’s data. Interpret the correlation coefficient.
If you use Excel, use Trendlines to find r2 = 0.9453. Round the square root to the nearest hundredth:
r = 0.97Since it is positive and greater than 0.75, there is a
strong positive correlation between the high temperature and the number of bottles sold.
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Find the correlation coefficient to the thousandth for the linear regression for the data in Check Your Understanding for Example 1. Interpret the correlation coefficient.
CHECK YOUR UNDERSTANDING
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Carlos entered data into his calculator and found a correlation coefficient of -0.28. Interpret this correlation coefficient.
EXTEND YOUR UNDERSTANDING
