Assignment 3
Question 1
Write a function that takes two matrices (say, A and B) as arguments and return the
matrix multiplication of A and B. Recall that in matrix multiplication, let C = A × B, then
Cij =P
k AikBkj . Notice that in matrix multiplication, the number of columns in A must
be the same as the number of rows in B. So in your program, if the inputs do not satisfy
the condition, then provide an error message. Notice that, you are not allowed to use the
R internal matrix multiplication function for this question, although you can verify your
solution with it.
Question 2
Suppose we had forgotten the formula for computing the area of a circle. All is not
lost. We can easily use the computer to approximate a circles area by simulation. For
convenience, we want to determine the area of the unit circle with radius 1 and name this
unknown constant π. We know that the area covered by a square with the corner points
(−1, 1), (1, 1), (1, −1), and (−1, −1) equals 4, as the length of each side is 2. This square
completely contains the unit circle. If we had a random point somewhere in the square,
we could calculate the probability that the point is also inside the unit circle. A random
point Z is determined by two coordinates, say X and Y , which have independent uniform
distributions on the interval [−1, 1]. The probability of Z being inside the circle is given by
the ratio of the circles area and the squares area or, more mathematically:
P(Z is inside the circle) = (circle area)/(square area) = π/4.
Knowing this, we can estimate (the unknown) π by multiplying the estimated probability
for Z being inside the circle by 4. Whats left is to generate a large number of random points
Z, calculate the proportion of points inside the circle, and multiply this by 4. This is our
estimate of π. Write a R program for this problem and estimate pi with 100, 1000, and 10,
000 random points. Try to work with complete vectors only and avoid loops. How far away
is the approximation from the pi we (fortunately) happen to know?
Hint: You need to know how to check if a point (x, y) is inside the circle. As the circle is
defined by its origin and the radius, the distance from the circles origin to the point is given.
Question 3
In statistical hypothesis testing, you specify a hypothesis about a population parameter
(your null hypothesis, or H0). You then draw a sample from this population and calculate
a statistic that’s used to make inferences about the population parameter. Assuming that
the null hypothesis is true, you calculate the probability of obtaining the observed sample
statistic or more extreme results. If the probability is sufficiently small, you reject the null
hypothesis in favor of its opposite (referred to as the alternative or research hypothesis, H1).
If the null hypothesis is false and the statistical test leads us to reject it, you’ve made
a correct decision. The power of a test is defined by the probability of rejecting the null
hypothesis given that the alternative hypothesis is correct.
If the null hypothesis is true and you don’t reject it, again you’ve made a correct decision.
If the null hypothesis is true but you reject it, youve committed a Type I error. If the null
hypothesis is false and you fail to reject it, youve committed a Type II error.
In this problem, we will use simulation to examine a statistical hypothesis test. We want
to test the null hypothesis that the population of mean of a normally distributed sample is
µ0 = 0, with known variance 1. Let the alternative be that the mean is equal to µ1 = 1. We
know that if we set the significance level to α (usually 1% or 5%) , the test will reject the
null hypothesis α% of the time (for a large number of samples), although the data actually
come from a normal distribution with mean 0 and variance 1 (the chance of making a type I
error is α% ). Show this by generating 1000 samples of size 100, calculating for each sample
whether or not the null hypothesis is rejected and summarizing how often the null hypothesis
was rejected (even when it is true). Recall that the criterion for rejecting the null hypothesis
in this case is that you reject the null hypothesis if
x¯ − µ0σ/√n> z1−α.
Question 4
This question is a continuation of Q3, in which we examine the power of the test.
• Estimate the power of your test in Q3 by generating 1000 samples of size 10 of the
population corresponding to the alternative hypothesis.
• Estimate the power of your test in Q3 by generating 1000 samples of size 10000 of the
population corresponding to the alternative hypothesis.

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