Bell Ringer CountryMarijuana (%) Other Drugs (%) Czech Rep.224 Denmark173 England4021 Finland51 Ireland3716 Italy198 Ireland2314 Norway63 Portugal73 Scotland5331 USA3424 A survey was conducted in the United States and 10 countries of Western Europe to determine the percentage of teenagers who had used marijuana and other drugs. The results are summarized in the table. 1)Look at the scatterplot and write a brief summary of the association. 2)What is the correlation between the percent of teens who have used marijuana and the percent who have used other drugs?

Linear Regression

One Double Whopper with cheese provides 53 grams of protein, 1020 calories, and 65 grams of fat. The correlation between Fat and Protein for 30 of the items on the Burger King menu is 0.83

If you want 25 grams of protein from a BK item, how much fat should you expect to consume? The association between protein and fat is a positive linear association with a fairly strong correlation of 0.83

Slopey-intercept

“Putting a hat on it” is notation to indicate that something has been predicted by a model.

Before using a regression model, we need to check the same conditions as we do for correlation: Quantitative variables Straight pattern Check for outliers

This model says that our predictions follow a straight line. The equation is given in slope-intercept form. For the association of fat and protein of Burger King items, the estimated linear model is:

What is the predicted amount of fat for the BK Broiler chicken sandwich, which has 30 grams of protein? Example

Moving one standard deviation away from the mean in one variable moves our estimate r standard deviations from the mean in the other variable. When the variables are standardized, the slope of the line turns out to be r.

Example A scatterplot of house price (in thousands of dollars) vs. house size (in thousands of square feet) for houses sold recently in Saratoga, NY shows a relationship that is straight, with only moderate scatter and no outliers. The correlation between Price and Size is )You go to an open house and find that the house is 1 standard deviation above the mean in size. What would you guess about its price? 2)You read an ad for a house priced 2 standard deviations below the mean. What would you guess about its size? 3)A friend tells you about a house whose size in square meters is 1.5 standard deviations above the mean. What would you guess about its size in square feet?

Answers 1)You should expect the price to be 0.77 standard deviations above the mean. 2)You should expect the size to be 2(0.77) = 1.54 standard deviations below the mean. 3)The home is 1.5 standard deviations above the mean in size no matter the units.

Residuals We predict that a BK Broiler chicken sandwich with 30 grams of protein should have 36 grams of fat, but it actually only has 25 grams of fat. The difference between the observed value and the predicted value is called the residual.

 A negative residual means the observed value is below the prediction on the regression line.  A positive residual means the observed value is above the prediction on the regression line.

If we keep the x-values and replace the y-values with the residuals, the resulting scatterplot has no pattern or direction.

Example 1)What does the slope of mean? 2)What are the units of the slope? 3)Your house is 2000 sq ft bigger than your neighbor’s house. How much more do you expect it to be worth? 4)Is the y-intercept of meaningful?

Answers 1)An increase in home size of 1000 sq ft is associated with an increase in price of $94,454 (or each 1 sq ft is associated with $94.45) 2)Units are in dollars per square foot 3)About $188,908 4)No… You can’t have a negative square ft. house

Example 1)Would you prefer to find a home with a negative or positive residual? Explain. 2)You plan to look for a home of about 3000 square feet. How much should you expect to have to pay? 3)You find a nice home that size which is selling for $300,000. What’s the residual?

Answers 1)Negative; that indicated it’s priced lower than a typical home of its size. 2)$280,250 3)$19,755

Today’s Assignment  Add to Homework: p. 192 #1-6, 11, 12