All of your work must be typeset using Rmarkdown and submitted online through the course’s blackboard shell.
Any work submitted late would be penalized as follows:
– any work submitted before midnight of the due date would attract a penalty of up to 10%
– any work submitted a day late would attract a penalty of up to 25%
– any work submitted two days late would attract a penalty of up to 50%
– otherwise you’ll be awarded a zero mark.
You are encouraged to discuss the assignment with others but at the end you must submit your individual work.
Any form cheating is not allowed and plagiarized work will be awarded a zero mark.
Q1. Examine the following code snippets and correct them where needed according to what they are intended for.
Please note that you can not a new code that does a similar task.
(a) Compute proportion estimate of obtaining a face of 1 or 2 in a fair die as
follows pˆ = Ixi≤2/n, where
i=1
Ixi≤2 = 1 xi ≤ 2
0 otherwise.
set. seed (1)
die = sample ( c(1:6) , 1000 , replace = TRUE ) success = rep (0 , 1000)
for( i in 1:100)
{
if( die > 2){ success[ i] == 1}
}
hat. p = MEAN ( success)
(b)
Q2. The Gini index is defined by the following equation.
2 Σn
ixi
n + 1
(a) Write down a user-defined function (call it gini.fn) that will calculate a value of this index for any given vector.
(b) Evaluate the Gini index for the number of hours spent watching television per week for a sample of 34 households.
23.1 15.9 21.0 26.0 25.1 14.7 24.2 16.6 18.2 16.5 20.7 15.3 17.7
19.1 22.7 21.9 14.6 26.3 25.8 9.4 17.0 21.2 17.9 24.7 21.1 17.2
19.1 22.7 24.0 24.7 22.5 8.3 2.5 30.4
Q3. A Newton method is one of the most popular numerical techniques used to finding roots of an algebraic function. That is, solving for f (x) = 0. According to the Newton method, if f (x) has a first derivative fJ(x) then the following algorithm will converge to a root of the above equation if the starting point is close enough.
a. Picking a starting value x0
b. For each estimate xn, calculate a new estimate
c. Repeat step (b) until the estimate are very close togetheror until the method fails
Use R and the Newton method to approximate the root of
HINT: A while loop is the most appropriate for this problem. But first use curve to visualize this graph and see where the root are.
. The Taylor series for sin−1(x) for |x| ≤ 1 is given by
(a) Write a function in R that will approximate sin−1(x) to some allowable toler- ance error. The arguments of your function should be x and tol (tolerance error), and return the approximate value of sin−1(x) and the total number of terms (N) summed together to give an approximate within the pre-stated tolerance error.
(b) Use your function to approximate sin−1(π/4) within a tolerance error of 0.001. Compare your solution with the R computed value.
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