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.

Answer : The relationship appear to be moderately strong and positive.

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

Answer : Explanatory variable is calories and response variable is carbohydrates

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

Answer : Because there is linear relationship between the two variables and we can make prediction from such a relationship

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

Answer : Yes the data meets the conditions

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. The scatterplot below shows the relationship between height and shoulder girth (over deltoid muscles), both measured in centimeters.

  1. Describe the relationship between shoulder girth and height.

Answer : It appears linear, and positively correlated.

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

Answer : The relationship wouldn’t change with change in centimeters to inches for girth. The plot would be more horizontally shrunk

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.

Answer : \[ he\hat{i}ght_i = 126.1503 + 0.4196798 * girth_i \]

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

Answer : On an average the height of a person in 126.1503 when shoulder girth is not considered. For each unit increase in shoulder girth increases the height by 0.4196798 times the shoulder girth.

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

Answer : This means around 45% of the variability in height can be explained by shoulder girth. 

R <- 0.67
R^2
## [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.

Answer : 

slope <- 0.67*(107.2/171.14)
estimated100 <- 126.1503 + slope*100
estimated100
## [1] 168.1183
  1. The student from part (d) is 160 cm tall. Calculate the residual, and explain what this residual means.

Answer : 160 - 168 = -8, negative residual means the model overestimated the height of the student

  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?

Answer : It would not be appropriate to use this model

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.

Answer : \[ hea\hat{r}twt_i= 4.034∗bodywt_i −0.357 \]

  1. Interpret the intercept.

Answer : Average heart weight will be -0.357 grams when body weight is 0.

  1. Interpret the slope.

Answer : For 1 unit increase in weight heart weight increase by 4.034 grams on average

  1. Interpret \(R^2\).

Answer : \(R^2\) = 64.66%, which means 64.66 % of the variability in heart weight is explained by body weight.

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

Answer : Slope is 0.1325028

# Given values
x <- -0.0883
y <- 3.9983
b0 <- 4.010

# Slope
b1 <- (y - b0) / x

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.

Answer : The p-value is so small that we reject the null hypothesis and conclude that beauty and teaching evaluation are not positively correlated.

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

Answer :

No, the conditions for linear regression are not met. It does not meet the condition of nearly-normal residuals.

Linearity: There does not appear to be a clear pattern of the residuals, suggesting that it may be due to a linear relationship.

Nearly normal residuals: The residuals apear to be slightly left-skewed, based on the histogram and normal plot.

Constant variablity: There appear to be roughly the same degree of residuals above and below the horizonal line, suggesting constant variability.