Description

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¯ =   x_i 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 =      ^x_i − 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