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**Least Squares Regression Line (LSRL)**

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Introduction Size of Diamond vs. Price Many times the scatterplot shows some pattern in the data. For now, we will look at the analysis of data that falls in a straight line pattern.

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Introduction When we see a straight line pattern, we want to model the data with a linear equation. This will allow us to make predictions and actually use our data.

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Linear Relations We know lines from algebra to come in the form y = mx + b, where m is the slope and b is the y-intercept. In statistics, we use y = a + bx for the equation of a straight line. Now a is the intercept and b is the slope. The slope (b) of the line, is the amount by which y increases when x increase by 1 unit. This interpretation is very important. The intercept (a), sometimes called the vertical intercept, is the height of the line when x = 0.

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**Example Consider the equation: y=7+3x The slope is 3.**

2 4 6 8 5 10 15 y = 7 + 3x a = 7 x increases by 1 y increases by b = 3 Consider the equation: y=7+3x The slope is 3. For every increase of 1 in the x-variable, there will be an increase of 3 in the y-variable. The intercept is 7. When the x-variable is 0, the y-variable is 7.

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**Example Consider the equation: y=17-4x The slope is -4.**

x increases by 1 y changes by b = -4 (i.e., changes by –4) 2 4 6 8 5 10 15 a = 17 Consider the equation: y=17-4x The slope is -4. For every increase of 1 in the x-variable, there will be a decrease of 4 in the y-variable. The intercept is 17. When the x-variable is 0, the y-variable is 17.

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**Make your own regression line**

Least Squares Line How can we find the best line to fit the data? We would like to minimize the total distance away from the line This distance is measured vertically from the point to the line. Go to the following applet and start plotting points to see how this process works. Make your own regression line

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**Least Squares Line You first get a line once you plot two points.**

When you plot the third, green bars appear representing the error (actually called residual) of the line. These are how far off your line is for each of the points. The best line is the one that would minimize the total length of the green lines (all put together).

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Guess the best fit line Go to the following applet to practice your skills at estimating an LSRL. Plot a bunch of points. Then click the draw line button and draw what you think is the best fit line Then, check the “show least squares line” checkbox To the applet

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The details of the LSRL The mathematics involved in calculating the LSRL is a bit complicated.

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Least Squares Line The most widely used criterion for measuring the goodness of fit of a line y = a + bx to bivariate data (x1, y1), (x2, y2), , (xn,yn) is the sum of the of the squared deviations about the line: The line that gives the best fit to the data is the one that minimizes this sum; it is called the least squares line or sample regression line.

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**Coefficients a and b The slope is: The intercept is:**

S-sub y and s-sub x are the sample standard deviations of y and x (kinda like rise over run) The slope is: The intercept is: y-bar and x-bar are the mean y and x respectively The equation of the least squares regression line is written as: The little symbol above the y is a hat! The equation is read as, “y-hat equals a plus bx.” The ‘y-hat’ indicates that this is a regression line and that the model (equation) is to be used to make predictions.

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**Three Important Questions**

To examine how useful or effective the line summarizing the relationship between x and y, we consider the following three questions. Is a line an appropriate way to summarize the relationship between the two variables? Are there any unusual aspects of the data set that we need to consider before proceeding to use the regression line to make predictions? If we decide that it is reasonable to use the regression line as a basis for prediction, how accurate can we expect predictions based on the regression line to be?

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**Example #1 - Finding the LSRL**

Consider the following data: With this data, find the LSRL Start by entering this data into list 1 and list 2 Shoe Size (men’s U.S.) Height (in) 7 64 10 69 12 71 8 68 9.5 10.5 70 11 72 12.5 74 13.5 77

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**Example #1 - Finding the LSRL**

You should then see the results of the regression. a=53.24 b=1.65 r-squared=.8422 r=.9177 This is the correlation coefficient for the scatterplot!!!

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**Example #2 – Interpreting LSRL**

Interpreting the intercept When your shoe size is 0, you should be about inches tall Of course this does not make much sense in the context of the problem Interpreting the slope For each increase of 1 in the shoe size, we would expect the height to increase by 1.65 inches

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**Example #3 – Using LSRL Making predictions**

How tall might you expect someone to be who has a shoe size of 12.5? Just plug in 12.5 for the shoe size above, so… Height = (12.5)= inches Of course this is a prediction and is therefore not exact.

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**Least Squares Regression Line (LSRL)**

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