Warm Up

1. Warm Up Ch.14

2. Find the following correlation. X is:1 5 8 12 16 Which has a mean of 8.4 and a standard deviation of 5.2. Y is: 1.5 3 8 9 15 Which has a mean of -7.3 and a standard deviation of 4.8. Interpret the correlation.

3. Which of the following correlations is not possible? Why? -1 1 .5 -.5 1.5

4. Chapter 15 Describing Relationships: Regression, Prediction and Causation

5. Regression lines If a scatterplot shows a linear relationship between two quantitative variables, we are often interested in summarizing the overall pattern by drawing a line on the graph. This type of line is called a regression line. A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We are often interested in using this line to predict a value of y for a given value of x.

6. Least-squares Regression Lines We want a line that comes closest to the points in the vertical direction. The most common is the least �squares method. The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. Note :We will not actually perform the least-squares method. Instead, we are interested in being able to use the resulting line.

7. Regression equations We write the regression equation in the form y = a + bx. y represents the response variable (on the y-axis) x represents the explanatory variable (on the x-axis) b represents the slope. b = r*(Sy/Sx) The slope tells us the amount by which y changes when x increases by one unit. In other words, b = 3 would mean that if the x variable increases one unit, then the y variable increases 3 units. a represents the y-intercept, the value of when x=0 a = ybar � b*xbar To use the equation for prediction, substitute in some x value and the equation will give you the resulting y value

8. The regression line between the age of a wife and the age of a husband is given by y=3.6+.97x where x is the wife�s age in years and y is the husband�s age in years. If a wife is 30 years old, how old is her husband? 27.8 32.5 32.7

9. Prediction Prediction is based on fitting some model to a set of data. We are looking at linear models, but there are also more elaborate models used for prediction Prediction works best when the model fits the data closely. The closer the data actually follows a pattern, the better the prediction will be. Prediction outside the range of the available data is risky. Height predictions � using a young child�s growth to predict how tall they will be at 25 Economic predictions � using the increase in gas prices over the past few months to estimate gas prices in 2007

10. Correlation and Regression correlation measures the strength and direction of a linear relationship. We now know that regression is what is used to draw the line representing this relationship. Correlation and regression are closely connected. Both correlation and regression are strongly affected by outliers. The usefulness of the regression line depends on the correlation between the two variables.

11. The Square of the correlation The square of the correlation, called R - squared. It is the proportion of the variation in the values of y that is explained by the least-squares regression of y on x. The idea is that when there is a straight-line relationship, some of the variation in y is accounted for by the fact that as x changes, it pulls y along with it. Ex. If r = .6, then = .36, meaning that roughly 36% of the variation is accounted for by the straight-line relationship. We use as the square of the correlation a measure of how successful the regression was in explaining the response.