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City Mileage, Highway Mileage. We expect a car's highway gas mileage to be related to its city gas mileage (in miles per gallon, mpg). Data for all 1259 vehicles in the government's 2019 Fuel Economy Guide give the regression line highway mpg \(=8.720+(0.914 \times\) city mpg \()\) for predicting highway mileage from city mileage. a. What is the slope of this line? Say in words what the numerical value of the slope tells you. b. What is the intercept? Explain why the value of the intercept is not statistically meaningful. c. Find the predicted highway mileage for a car that gets \(16 \mathrm{mpg}\) in the city. Do the same for a car with city mileage of \(28 \mathrm{mpg}\). d. Draw a graph of the regression line for city mileages between 10 and \(50 \mathrm{mpg}\). (Be sure to show the scales for the \(x\) and \(y\) axes.)

Homicide and Suicide. Preventing suicide is a important issue facing mental health workers. Predicting geographic regions where the risk of suicide is high could help people decide where to increase or improve mental health resources and care. Some psychiatrists have argued that homicide and suicide may have some causes in common. If so, one would expect homicide and suicide rates to be correlated. And if this is true, areas with high rates of homicide might be predicted to have high rates of suicide and therefore be in need of increased mental health resources. Research has had mixed results, including some evidence that there is a positive correlation in certain European countries but not in the United States. Here are data from 2015 for the 11 counties in Ohio with sufficient data for homicides and suicides to allow for estimating rates for both. \(\underline{\text { Rates }}\) are per 100,000 people. DEATH a. Make a scatterplot that shows how suicide rate can be predicted from homicide rate. There is a weak linear relationship, with correlation \(r=-0.0645\). b. Find the least-squares regression line for predicting suicide rate from homicide rate. Add this line to your scatterplot. c. Explain in words what the slope of the regression line tells us. d. Another Ohio county has a homicide rate of \(8.0\) per 100,000 people. What is the county's predicted suicide rate?

The Price of Diamond Rings. Online advertisements contained pictures of diamond rings and listed their prices, diamond weights (in carats), and gold purity. Based on data for only the 18-carat gold ladies' rings in the advertisements, the least-squares regression line for predicting price (in dollars) from the weight of the diamond (in carats) is \(\frac{18}{18}\) $$ \text { price }=-6047.75+11975.14 \text { carats } $$ a. What does the slope of this line say about the relationship between price and number of carats? b. What is the predicted price when number of carats \(=0\) ? How would you interpret this price?

What's My Grade? In Professor Krugman's economics course, the correlation between the students' total scores prior to the final examination and their final-examination scores is \(r=0.5\). The pre-exam totals for all students in the course have mean 280 and standard deviation 40 . The finalexam scores have mean 75 and standard deviation 8 . Professor Krugman has lost Julie's final exam but knows that her total before the exam was 300 . He decides to predict her final-exam score from her pre-exam total. a. What is the slope of the least-squares regression line of final-exam scores on pre-exam total scores in this course? What is the intercept? Interpret the slope in the context of the problem. b. Use the regression line to predict Julie's final-exam score. c. Julie doesn't think this method accurately predicts how well she did on the final exam. Use \(r^{2}\) to argue that her actual score could have been much higher (or much lower) than the predicted value.