library(DATA606)

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. 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.

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

The relationship is linear in that increased in calories has an effect of increased in carbs

  1. In this scenario, what are the explanatory and response variables?

Explanatory: Calories. Response: Carbs

  1. Why might we want to fit a regression line to these data?

One reason may be to predict/estimate/monitor carbs given calories

  1. Do these data meet the conditions required for fitting a least squares line?

Partly. Based on residual plot and histogram, it nearly meets linearity and normality, and independent observation can be assumed. However, it seems to fail the constant variability condition

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.

  1. Write the equation of the regression line for predicting height.
R <- 0.67
ybar <- 171.14
xbar <- 107.20
sy <- 9.41
sx <- 10.37

slope <- R * sy / sx
b0 <- ybar - xbar*slope

height = 105.9650878 + 0.6079749 * shoulder girth

  1. Interpret the slope and the intercept in this context.

Each centimeter in shoulder girth predicts an additional 0.6079749 cm to the height

  1. Calculate R2 of the regression line for predicting height from shoulder girth, and interpret it in the context of the application.

R2 is 0.4489, the proportion of the variability in height that is explained by the shoulder girth

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

Plugging the shoulder girth to the model, the estimate height is 166.7625805

  1. The student from part (d) is 160 cm tall. Calculate the residual, and explain what this residual means.

Residual between actual and estimate is -6.7625805 cm, a negative residual. The model overestimates the height

  1. A one year old has a shoulder girth of 56 cm. Would it be appropriate to use this linear model to predict the height of this child?

No, it would be extrapolation

7.30 Cats, Part I.

The following regression output is for predicting the heart weight (in g) of cats from their body weight (in kg). The coeffcients are estimated using a dataset of 144 domestic cats.

  1. Write out the linear model.

heart’s weigh = -0.357 + 4.034 * body weight

  1. Interpret the intercept.

Expected heart’s weight with 0 body weight is -0.357 kg which does not make sense, but just serves to adjust the height of the regression line

  1. Interpret the slope.

For each increase in body’s weight in kg, we predict an increase of heart’s weight by 4.034 g

  1. Interpret R2.

Body’s weight explains 64.66% of the variablility in heart’s weight

  1. Calculate the correlation coefficient.

0.8041144

7.40 Rate my professor.

Many college courses conclude by giving students the opportunity to evaluate the course and the instructor anonymously. However, the use of these student evalu- ations as an indicator of course quality and teaching effectiveness is often criticized because these measures may reflect the influence of non-teaching related characteristics, such as the physical appearance of the instructor. Researchers at University of Texas, Austin collected data on teaching evaluation score (higher score means better) and standardized beauty score (a score of 0 means average, negative score means below average, and a positive score means above average) for a sample of 463 professors. The scatterplot below shows the relationship between these variables, and also provided is a regression output for predicting teaching evaluation score from beauty score.

  1. Given that the average standardized beauty score is -0.0883 and average teaching evaluation score is 3.9983, calculate the slope. Alternatively, the slope may be computed using just the information provided in the model summary table.
ybar <- 3.9983
xbar <- -0.0883
b0 <- 4.010
slope <- (ybar - b0) / xbar; slope  
## [1] 0.1325028
  1. Do these data provide convincing evidence that the slope of the relationship between teaching evaluation and beauty is positive? Explain your reasoning.

Yes the slope is positive, although this seems to be very weak relationship

  1. List the conditions required for linear regression and check if each one is satisfied for this model based on the following diagnostic plots.

Linearity there seems to be somewhat linear, but not strong relationship

Nearly normal residuals historgram seems nearly normal with negative skew

Constant variability residual plot seems variability is roughly constant

Independent observations could not be exactly determined, could be assumed