Describe the relationship between number of calories and amount of carbohydrates (in grams) that Starbucks food menu items contain.

7.22 Over-under, Part II. Suppose we fit a regression line to predict the number of incidents of skin cancer per 1,000 people from the number of sunny days in a year. For a particular year, we predict the incidence of skin cancer to be 1.5 per 1,000 people, and the residual for this year is 0.5. Did we over or under estimate the incidence of skin cancer? Explain your reasoning.

7.24 Nutrition at Starbucks, Part I. The scatterplot below shows the relationship between the number of calories and amount of carbohydrates (in grams) Starbucks food menu items contain.21 Since Starbucks only lists the number of calories on the display items, we are interested in predicting the amount of carbs a menu item has based on its calorie content.

(a) Describe the relationship between number of calories and amount of carbohydrates (in grams) that Starbucks food menu items contain.

(b) In this scenario, what are the explanatory and response variables?

(c) Why might we want to fit a regression line to these data?

(d) Do these data meet the conditions required for fitting a least squares line?

7.26 Body measurements, Part III. Exercise 7.15 introduces data on shoulder girth and height of a group of individuals. The mean shoulder girth is 107.20 cm with a standard deviation of 10.37 cm. The mean height is 171.14 cm with a standard deviation of 9.41 cm. The correlation between height and shoulder girth is 0.67.

(a) Write the equation of the regression line for predicting height.

(b) Interpret the slope and the intercept in this context.

(c) Calculate R 2 of the regression line for predicting height from shoulder girth, and interpret it in the context of the application.

(d) A randomly selected student from your class has a shoulder girth of 100 cm. Predict the height of this student using the model.

7.28 Helmets and lunches. The scatterplot shows the relationship between socioeconomic status measured as the percentage of children in a neighborhood receiving reduced-fee lunches at school (lunch) and the percentage of bike riders in the neighborhood wearing helmets (helmet). The average percentage of children receiving reduced-fee lunches is 30.8% with a standard deviation of 26.7% and the average percentage of bike riders wearing helmets is 38.8% with a standard deviation of 16.9%.

(a) If the R 2 for the least-squares regression line for these data is 72%, what is the correlation between lunch and helmet?

(b) Calculate the slope and intercept for the leastsquares regression line for these data.

(c) Interpret the intercept of the least-squares regression line in the context of the application.

(d) Interpret the slope of the least-squares regression line in the context of the application.

7.38 Husbands and wives, Part III. Exercise 7.37 presents a scatterplot displaying the relationship between husbands’ and wives’ ages in a random sample of 170 married couples in Britain, where both partners’ ages are below 65 years. Given below is summary output of the least squares fit for predicting wife’s age from husband’s age.

(b) Write the equation of the regression line for predicting wife’s age from husband’s age.

(c) Interpret the slope and intercept in context.

(d) Given that R 2 = 0.88, what is the correlation of ages in this data set?

(e) You meet a married man from Britain who is 55 years old. What would you predict his wife’s age to be? How reliable is this prediction?

Open the dataset BodyFat.RData in R. Review the R code for correlation and linear regression. This problem is worth 5 points.

You are a physician working at a clinic and are concerned with the rising level of obesity amongst your patients. With the help of some colleagues, you obtain some patient data from their charts and compile it as a dataset on R (BodyFat.RData.)

While in a meeting, a debate arises over the best predictor of patient body fat percentage. Dr. Bob believes that Age (measured in years) is the best indicator of a patient’s body fat. Dr. John believes that a person’s Weight (measured in pounds) is the most appropriate variable. You however, are certain that a patient’s Neck circumference (measured in inches) would most accurately predict a person’s body fat percentage.

You whip out your laptop and open R Studio to settle this debate:

1. Create scatterplots with regression lines for Bodyfat predicted by: 1) Age 2) Weight 3) Neck circumference. From the scatterplots, describe each relationship as weak, moderate, or strong.
2. Calculate the R2 for each of the regression models. Which linear regression model has the best fit? Which Doctor is correct? Explain your answer using your R-generated data.
3. You are a 35 year old 5 foot 10-inch tall and 185 pound male with a neck circumference of 38 inches. Using the best fitted regression model, predict your body fat percentage. Show your regression formula in y=mx+b format.
4. Is the regression model you chose a good fit to explain body fat percentage? Can you think of a reason why your model may not be a good predictor for body fat?
5. Is Body Mass Index (BMI) a better predictor for body fat percentage? Show a scatterplot with regression line and report R2. From your thorough analysis, pick the most appropriate variable for predicting body fat percentages in your clinic.