Nutrition at Starbucks, Part I. (8.22, p. 326) 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.

Its is a positive, linear association between the number of calories and amount of carbohydrates i.e increase in calories corresponds to increase in carbs but variation in residuals increases as calories increase.

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

explanatory variables: calories response variables: carbs

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

Since we want to predict carbs for given number of calories using regression line.

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

Body measurements, Part I. (8.13, p. 316) Researchers studying anthropometry collected body girth measurements and skeletal diameter measurements, as well as age, weight, height and gender for 507 physically active individuals.19 The scatterplot below shows the relationship between height and shoulder girth (over deltoid muscles), both measured in centimeters.

\begin{center}

\end{center}

  1. Describe the relationship between shoulder girth and height.

Its is a positive, linear association between shoulder girth and height.

  1. How would the relationship change if shoulder girth was measured in inches while the units of height remained in centimeters?

The relationship would become more linear and strong since the residuals become closer.

Body measurements, Part III. (8.24, p. 326) Exercise above 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.

\(\widehat{y}\) = \(\beta_0\) + \(\beta_1\) * x

\(\beta_1\) = \(\frac{S_y}{S_x}\) * R

s_y <- 9.41
s_x <- 10.37
r <- 0.67
y_mean <- 171.14
x_mean <- 107.20
beta1 <- (s_y / s_x ) *r
beta0 <- y_mean - (beta1*x_mean)
cat(beta0, beta1)
## 105.9651 0.6079749

Equation of regression line is

\(\widehat{y}\) = 105.9651 + 0.6079749*x

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

slope: For each increase in shoulder girth, model predicts 0.61cm additional in height. intercept: The intercept represents the height in cm when shoulder girth is 0cm.

  1. Calculate \(R^2\) of the regression line for predicting height from shoulder girth, and interpret it in the context of the application.
r_square <- r^2
r_square
## [1] 0.4489

It shows that the linear model explains 44.89% variation of height data.

  1. A randomly selected student from your class has a shoulder girth of 100 cm. Predict the height of this student using the model.
ht <- 105.9651 + 0.6079749*100
ht
## [1] 166.7626
  1. The student from part (d) is 160 cm tall. Calculate the residual, and explain what this residual means.
res <- 160 - ht
res
## [1] -6.76259

A negative residual means our model overestimates the height data.

  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?

The original dataset has shoulder girth between 85cm to 135cm so this linear model would NOT be appropriate for value 56cm as it is out of range.

Cats, Part I. (8.26, p. 327) 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.

\begin{center} \end{center}

  1. Write out the linear model.

\(\widehat{y}\) = -0.357 + 4.034 * x

  1. Interpret the intercept.

Here intercept is a negative value -0.357 which means heart weight will be -0.357 when weight is 0 kg.

  1. Interpret the slope.

Here is the slope is 4.034 which means heart weight increases by 4.034 grams for each 1kg increase in cats body weight.

  1. Interpret \(R^2\).

It shows that this linear model explains 64.66% variation in heart weight.

  1. Calculate the correlation coefficient.
R2 <- 0.6466
R <- sqrt(R2)
R
## [1] 0.8041144

Rate my professor. (8.44, p. 340) 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.

\begin{center}

\end{center}

  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.

Equation: \(\widehat{y}\) = \(\beta_0\) + \(\beta_1\) * x

slope <- (4.010 - 3.9983) / 0.0883
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.

\(\beta_1\) = \(\frac{S_y}{S_x}\) * R

Here \(S_x\) and \(S_y\) are positive so the data shows slope 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: The relationship between beauty and teaching evaluation seems linear from the scatter plot.

Nearly normal residuals: As shown in the residuals histogram, they are nearly normal.

Constant variability: The scatterplot of the residuals shows constant variability.

Independent observations: I assume data is independent in the sample of 463 professors.