Chapter 3: Linear models EQ: How can bivariate data be analyzed?

Scatter Plots can SAVE lives The forecasted temperature for January 20, 1986 was 31 degrees.

Scatterplots Shows the relationship between 2 quantitative variables. X: explanatory variable Y: response variable

Analysis Look at the overall form and any deviations Form: Linear or not linear and does the Data have any Clusters, outliers, or patterns Direction: Positive or negative association Strength: weak, moderate, strong

Examples Explanatory Variable Number of bikes Response Variable Number of accidents Form: Linear. Possible outlier Direction: Positive association Strength Strong

Examples Explanatory Variable Time swimming Response Variable Pulse rate Form: Linear. Scatter seems to increase as the time increases Direction: Negative association Strength moderate

Examples Explanatory Variable Percent without HS diploma Response Variable Teacher Pay Form: Large cluster of data between 10 and 30 %. Direction: none Strength weak

Example Effects of Humor on Test Anxiety and Performance Anxiety Score 23 43 14 59 48 77 7 50 20 52 46 15 51 21 Construct a scatterplot of anxiety vs. exam score. Describe the strength, direction, and form.

Correlation Definition: The strength of the linear relationship Notation: Correlation = r Values -1 < r < 1 Interpretation The closer r is to 1 or -1 indicates a stronger linear relationship A correlation near 0 indicates no linear relationship Cautions: Look at the data first and make sure a linear relationship is feasible

Examples

Line of best fit Method: Minimize the sum of the squared errors Prediction line Predicted value Error Actual value

LSRL Equation Y-intercept Y-hat Slope

Facts about the LSRL Line Every LSRL line passes through All found in 2 variable stats

Interpretations Slope: For every 1 unit increase in x, the y changes by the slope Intercept: When x is 0, y is a. May not have any PRACTICAL meaning

Dinosaur Bones Archeologists want to determine if a new bone belongs to a certain species of dinosaur. They have a set of bones that they KNOW go together and have recorded the Femur lengths and Humerus lengths. Analyze the data and determine if there is a relationship. Create a scatterplot Analyze the scatterplot Find the correlation Find the LSRL Interpret Femur Humerus 38 41 56 63 59 70 64 72 74 84