EQ: How well does the line fit the data? Analysis of LSRL EQ: How well does the line fit the data?

What would you conclude based on the graph? Data from 1860-1940 Barrels of Rum Sold Ministers in Boston

Reasons for strong correlation Lurking variables: Something in the background that affects both variables the same way. For the historical data, as population increased the number of ministers increased as did the amount of alcohol being sold.

Shoe size vs. Height Determine if a linear model is a good idea. 8.5 66 9 68.5 67.5 9.5 70 10 72 10.5 71.5 69.5 11 73 12 73.5 74 12.5 Determine if a linear model is a good idea. Create a scatterplot and examine the relationship. DON’T FIND THE LSRL!!!!

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

Find the LSRL using summary statistics

Interpret all of the values r : There is a strong positive linear relationship a: when the shoe size is 0 the height is predicted to be 51.36 inches b: for every 1 increase in shoe size the height will increase by 1.87 inches

Coefficient of Determination The percent of variation in the y values that is explained by the linear model with x. Coefficient of determination = r2 where r is the correlation.

The coefficient of determination between shoe size and height is. 8575 The coefficient of determination between shoe size and height is .8575. What does this mean???? 85.75% of the variation in heights is explained by the linear model with shoe size.

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

Residuals A residual is the difference between an observed value of the response variable and the value predicted by the LSRL

Predictions Predict the height of a person with a shoe size of 8.5

Residual Actual Data: Residual: Observed - Predicted 66-67.25 =-1.25 8.5 shoe size and 66 inches for height 66-67.25 =-1.25

Create the LSRL

Calculate Residuals

Evaluate Residual Plot A good residual plot has No patterns No outliers Balance between +’s and –’s

Bad Residual Plots

Example How are American female (30-39) heights and weights related? Create a scatterplot and comment on the relationship Determine the LSRL, r, and r2 and interpret the values. Evaluate the model by analyzing the residuals. Predict the weight for a 63 inch female. Calculate the residual for a 63 inch female. How confident are you in your prediction?