STAT823- Homework 4 Solved

Directions

The aim of this exercise is to make you get used to using R and learn how to write a dynamic report. Explore the basic functionalities of R and include enough documentation with annotated comments.

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Show all your work/code. Late submission will attract a penalty of 10 points per day after the due date.

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1. A total of 30 Masters students in a Statistics course were asked to state how long they take to finish their weekly homework.

The following is the reported time in hours:

2,3,1.5,3.5,4,3,1,3,2,3.5,4,3,2.5,2.5,2,2,1.5,2,2,3,3,

3,1,1,1,1.5,2,2.5,2.5,3

(a) Write your own R code (function) to calculate the sample mean without using the mean() R function then confirm

your answer using the mean() function

1 Xn

x¯ =    xi n i=1

(b) Write your own R code (function) to calculate the sample variance without using the sd() and var() R functions. You can use the mean() and sum() functions, then confirm your answer using the var() function

                                                                                                        1  n                             

2                                                    X 2          2
s =       xi − nx¯  n − 1 i=1

2. Caleb wishes to take P = $10,000 loan from a bank. The bank offers the loan at a monthly interest rate r = 3% for a period of n = 24 months. Calculate the monthly instalments m that Caleb will have to remit to the bank given that the principal is calculated as

1 − (1 + r)−n

                                                                            P = m    

r

(Hint: First, make m the subject of the formula. That is, re-write the expression to obtain m = (.) then solve for m)

r

m = P ∗ 

1 − (1 + r)−n

 

Proof: Not reqired

                                                          2                                  1 Xn                         2

S                      =           (xi − x¯)  n − 1 i=1

                                                                                                         1 Xn          2                             2 

=   (xi − 2xix¯ + ¯x ) n − 1 i=1

                                                                                                         1 Xn         2                Xn                     2

=   xi − 2x¯ xi + nx¯  n − 1 i=1 i=1

                 Notice that    ¯         =               1 Xn Xi ⇒ Xn Xi = nX¯

X n i=1                i=1

                                                                                                          1  n                                               

                                                                    ∴ S2 =         X x                                 

n − 1 i=1

                                                                                                          1 Xn                                              

=           x       n − 1 i=1



 =            n − 1 i=1