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.

It appears as though the more calories a drink has, the more grams of carbohydrates the drink has as well.

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

Explanatory = Calories Response = Carbs (grams)

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

Fitting a regression line to this data would allow us to comprehend the relationship between these variables.

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

It appears as though the residuals are relatively normally distributed, therefore we can assume the conditions are 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.

  1. Write the equation of the regression line for predicting height.

Shoulder Girth = explanatory variable (x) Height = response variable (y)

#height stats
sy <- 9.41
y_bar <- 171.14
#shoulder stats
sx <- 10.37
x_bar <- 107.20

r <- 0.67

slope <- r * sy / sx
b0 <- y_bar - x_bar*slope
slope
## [1] 0.6079749
b0
## [1] 105.9651

height = 105.9651 + (shoulder girth)*(.607)

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

If shoulder girth is 0, we expect the height to be 105.9651 cm, and with each inch increase of shoulder girth, we expect height to increase 0.6079749 cm.

  1. Calculate R2 of the regression line for predicting height from shoulder girth, and interpret it in the context of the application.
r*r
## [1] 0.4489
  1. A randomly selected student from your class has a shoulder girth of 100 cm. Predict the height of this student using the model.
height <- 105.9651 + (100)*(.607)
height
## [1] 166.6651
  1. The student from part (d) is 160 cm tall. Calculate the residual, and explain what this residual means.

The residual is the difference between the actual height and the predicted heigh of the student.

residual <- 166.6651 - 160
residual
## [1] 6.6651
  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?

It would be innappropriate to be using this linear model because we would be extrapolating on our dataset to include children.

height <- 105.9651 + (57)*(.607)
height
## [1] 140.5641

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 coefficients are estimated using a dataset of 144 domestic cats.

  1. Write out the linear model.

explanatory = body weight response = heart weight

  1. Interpret the intercept.

heart weight = -0.357 + (body weight)*(4.034)

  1. Interpret the slope.

As the body weight increases by 1 gram, the heart weight increases by 4.034 grams.

  1. Interpret R2.

The R^2 = 64.66%, which demonstrates the relationship between the two variables. It shows that they are correlated by 64.66%

  1. Calculate the correlation coefficient.
r2 <- .6466
sqrt(r2)
## [1] 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 evaluations 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.

teaching = yintercept + B1*beauty

avg_beauty <- -0.0883
avg_teaching <- 3.9983
y_int <- 4.010

B1 <- (avg_teaching - y_int)/avg_beauty
B1
## [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.

Since the regreesion model has a p value that is less than .05, there is convincing evidence that the slope of the relationship between teaching evaluation and beauty is positive

  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 is satisfied. The data shows a linear trend.

Nearly normal residuals is satisified. The residuals appear to be randomized and follow a normal distribution.

Constant variability is satisified. The variability of points around the least squares line remains roughly constant.

Independent observations is assumed.