3. Measuring body fat is not simple. Muscle and bone are denser than fat so an estimate of body density can be used to estimate the proportion of fat in the body. Measuring someone’s weight is easy but volume is more difficult. One method requires submerging the body underwater in a tank and measuring the increase in the water level. Most people would prefer not to be submerged underwater to get a measure of body fat so we would like to have an easier method. In order to develop such a method, researchers recorded age, weight, height, and 10 body circumference measurements for 252 men. Each man’s percentage of body fat was accurately estimated by an underwater weighing technique. We wish to predict body fat using just the easy-to-record measurements. For simplicity, four variables selected for the study are listed in Table.

Use data: Project-2-data.txt

Variable         Description

X1         chest circumference (cm)

X2         percent body fat using Brozek’s equation

X3         weight (lbs) (‘Heavy’ if greater than 180, ‘Light’ otherwise)

X4         height (inches) (‘Tall’ if greater than 70, ‘Short’ otherwise)

(a) For each variable, describe it as quantitative or qualitative.

(b) Develop appropriate descriptive statistics to summarize the data.

(c) Draw a histogram for each continuous variable. Interpret.

(d) Draw a scatter plot between two continuous variables.

(e) Draw a bar graph for each categorical variable.

(f) For each continuous variable, compute the 95% confidence interval for a population mean.

(g) For each continuous variable, compute the 95% confidence interval for a population variance.

(h) For each continuous variable, perform a statistical test to see whether its population mean is zero or not.

(i) For each categorical variable, perform a t-test on x2. (that is, test whether there is significant mean difference on x2 between two levels of each categorical variable)

(j) Calculate correlations between continuous variables.

(k) Perform a statistical test to see whether the correlation between two continuous variables is significant or not.

(l) Test independence between categorical variables.

(m) Fit a simple linear regression model relating x2 to x1 (that is, x2 is a response variable). Write down the fitted model, and interpret.