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.
- Describe the relationship between number of calories and amount of carbohydrates (in grams) that Starbucks food menu items contain.
Answer: We can see that relationship between the amount of carbonhydrates and the number of calories has a strong positive correlation. In other words, as the number of calories increases, the amount of carbonhydrate will likely to increase as well.
- In this scenario, what are the explanatory and response variables?
Answer: The explanatory variable will be number of calories and the response variable will be amount of carbonhydrates.
- Why might we want to fit a regression line to these data?
Answer: We might want to fit a regression line to these data because there is a linear relationship between the two variables and can be used to predict the expected amount of carbonhydrates present in the Starbucks menu items based on its number of calaries.
- Do these data meet the conditions required for fitting a least squares line?
Answer: Yes. These data meet the conditions required for fitting a least squares line because we can see that the points in the residual plot are randomly dispersed around the horizontal dashed line at y = 0 and does not show any specific pattern such as U-shaped and inverted-U which better fit for a non-linear model.So least squares line can be fitted to these data.
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.
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- Describe the relationship between shoulder girth and height.
Answer: There is a strong positive linear relationship between the two variables, shoulder girth and height. The positive linear trend of the points shows that as the shoulder girth increases, the height will likely to increase as well.
- How would the relationship change if shoulder girth was measured in inches while the units of height remained in centimeters?
Answer: There will be no change in the relationship between shoulder girth and height even though their units measurement is different because relationship is invariant from units of variables. Thus, change the units of the height or shoulder girth will not have any effect on linear relationship between the two variables. The linear relationship will remain the same.
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.
- Write the equation of the regression line for predicting height.
Answer: The linear equation is y=a+bx. Slope, b = 0.67 * (9.41/10.37) = 0.608, Intercept, a = y - bx = 171.14 - 0.608*107.2 = 105.96. Therefore, the equation of the regression line for predicting height is y = 105.96 + 0.608x.
- Interpret the slope and the intercept in this context.
Answer: Intercept: the intercept 105.96 is the height when shoulder girth is 0. Slope: for each unit in cm increase in shoulder girth, the height increases by 0.608 cm.
- Calculate \(R^2\) of the regression line for predicting height from shoulder girth, and interpret it in the context of the application.
Answer: \(R^2\) = 0.67^2 = 0.4489 or 44.89%. It means that 44.89% of the variation in height is explained by the variation in shoulder girth.
- A randomly selected student from your class has a shoulder girth of 100 cm. Predict the height of this student using the model.
Answer: y = 105.96 + 0.608 * (100) = 166.76. Therefore, the model predicts that the height of this student is 166.76 cm.
- The student from part (d) is 160 cm tall. Calculate the residual, and explain what this residual means.
Answer: Residual = Observed - Predicted. So, Residual = 160 - 166.76 = -6.76. This means that the model prediction is overestimated by 6.76 cm.
- 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: The original data set shoulder girth variable values are between 80 cm and 140 cm. Therefore, 56 cm is outside the range of the sample data. Therefore,it would not be appropriate to use this linear model to predict the height of this child.
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.
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- Write out the linear model.
Answer: The linear model is Predicted Heart Weight = -0.357 + 4.034*(Body weight)
- Interpret the intercept.
Answer: The intercept (constant value) is the expected mean value of the predicted heart weight when body weight equals to zero. In our case, when we set the body weight equals to zero, the predicted heart weight is -0.357 kg. But this predicted value is worthless because weight cannot be negative.
- Interpret the slope.
Answer: Slope is the measure of the steepness of a line. It is calculated by finding the ratio of the vertical change to the horizontal change between two distinct points on a line. Here the slope is 4.034. It means keeping everything else fixed if we increase the body weight by 1 unit then the predicted heart weight will be increased by 4.034.
- Interpret \(R^2\).
Answer: \(R^2\) =64.66%. It means that 64.66% of the variation in predicted heart weight is explained by the variation in body weight. The 64.66% seems to be medium goodness of fit of regression line and is a fairly good model.
- Calculate the correlation coefficient.
Answer: The correlation coefficient is simply the square root of R^2 = SQRT(0.6466) = 0.80.
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.
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- 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: The slope can be calculated based on information from summary table, slope = t- value of beauty * standard error of beauty = 4.13*0.0322 = 0.133. Or we can use the linear equation, y=a+bx. So, it will be 3.9983 = 4.010 + b(-0.0883) and b is the slope. Rearrange the model, b = (3.9983-4.010)/-0.0883 = 0.1325 = 0.133. Therefore, the slope is 0.133.
- Do these data provide convincing evidence that the slope of the relationship between teaching evaluation and beauty is positive? Explain your reasoning.
Answer: Yes. These data provide convincing evidence that the slope of the relationship between teaching evaluation and beauty is positive because both p-values for the intercept and slope are statistically significant.
- List the conditions required for linear regression and check if each one is satisfied for this model based on the following diagnostic plots.