Chapter Wrap Up
Concept Check
Section Reviews
9.1 Introduction to Bi-variate Data and Scatterplots
Scatter plots are particularly helpful graphs when we want to see if there is a linear relationship among data points. They indicate both the direction of the relationship between the x variables and the y variables, and the strength of the relationship. We calculate the strength of the relationship between an independent variable and a dependent variable using linear regression.
9.2 Measures of Association
The correlation coefficient r measures the strength of the linear association between x and y. The variable r has to be between –1 and +1. When r is positive, the x and y will tend to increase and decrease together. When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. The coefficient of determination r2, is equal to the square of the correlation coefficient. When expressed as a percent, r2 represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression line.
9.3 Modeling Linear Relationships
A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called “errors,” measure the distance from the actual value of y and the estimated value of y. The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data.
9.4 Cautions about Regression
To determine if a point is an outlier, do one of the following:
- Must have a linear relationship to use these methods!
- Correlation is not causation
- Be careful with Extrapolation
- Beware of Influential Points
9.5 Inference for Regression
We can apply our inference techniques to regression, especially for the slope.
Key Terms
Try to define the terms below on your own. Scroll over any term to check your response!
9.1 Introduction to Bi-variate Data and Scatterplots
9.2 Measures of Association
9.3 Modeling Linear Relationships
9.4 Cautions about Regression
Extra Practice
9.1 Introduction to Bi-variate Data and Scatterplots
1. The Gross Domestic Product Purchasing Power Parity is an indication of a country’s currency value compared to another country. The figure below shows the GDP PPP of Cuba as compared to US dollars. Construct a scatter plot of the data.
| Year | Cuba’s PPP | Year | Cuba’s PPP |
|---|---|---|---|
| 1999 | 1,700 | 2006 | 4,000 |
| 2000 | 1,700 | 2007 | 11,000 |
| 2002 | 2,300 | 2008 | 9,500 |
| 2003 | 2,900 | 2009 | 9,700 |
| 2004 | 3,000 | 2010 | 9,900 |
| 2005 | 3,500 |
2. The following table shows the poverty