[Solved] MA331 Homework 01

Problem 1. Complete the following problems for the sample of two variables (X,Y ) by using R.

(0.2,1.1),(1.2,2.3),(0.9,1.1),(2.2,3.6),(3.2,0.1),(0.3,1.0),(1.7,6.9)

(3.1,4.8),(2.3,6.5),(1.5,7.8),(2.5,5.8),(3.0,8.0),(2.6,9.4),(9.0,9.8).

• Plot the histogram, pie chart of X and Y , respectively, and then describe their distributions. (Hint: Since histogram and pie chart are for categorical variables, numerical observation ought to be grouped before applying the two functions.)
• For X and Y , build the box-plot, compute their five-number summaries and variances, respectively. Are there any outliers of X and Y , respectively?
• Produce the scatter plot of (X,Y ) and evaluate their correlation coefficient. Qualitatively describe the linear association between X and Y .
• Are there any outliers of (X,Y )? If yes, remove them and compute the correlation coefficient again.
• What difference do you observed between the numerical results in (iii) and (iv)?
• Produce the normal QQ plots for observations of X and Y , respectively, and then determine which one is more likely to be of normal distribution?

Problem 2. For x₁, ,x_n, show that

and thus

.