Andrew ng ML Coursera Quiz1

1. 
   Consider the problem of predicting how well a student does in her second year of college/university, given how well they did in their first year.
   Specifically, let x be equal to the number of “A” grades (including A-. A and A+ grades) that a student receives in their first year of college (freshmen year). We would like to predict the value of y, which we define as the number of “A” grades they get in their second year (sophomore year).
   Questions 1 through 4 will use the following training set of a small sample of different students’ performances. Here each row is one training example. Recall that in linear regression, our hypothesis is hθ(x)=θ0+θ1x, and we use m to denote the number of training examples.

For the training set given above, what is the value of m? In the box below, please enter your answer (which should be a number between 0 and 10).

1. 
   Consider the following training set of m=4 training examples:
   x y
   1 0.5
   2 1
   4 2
   0 0
   Consider the linear regression model hθ(x)=θ0+θ1x. What are the values of θ0 and θ1 that you would expect to obtain upon running gradient descent on this model? (Linear regression will be able to fit this data perfectly.)

θ0=0.5,θ1=0.5

θ0=0,θ1=0.5

θ0=1,θ1=1

θ0=1,θ1=0.5

θ0=0.5,θ1=0
3.
Suppose we set θ0=−1,θ1=2. What is hθ(6)?

1. 
   Let f be some function so that
   f(θ0,θ1) outputs a number. For this problem,
   f is some arbitrary/unknown smooth function (not necessarily the
   cost function of linear regression, so f may have local optima).
   Suppose we use gradient descent to try to minimize f(θ0,θ1)
   as a function of θ0 and θ1. Which of the
   following statements are true? (Check all that apply.)

No matter how θ0 and θ1 are initialized, so long
as α is sufficiently small, we can safely expect gradient descent to converge
to the same solution.

Setting the learning rate α to be very small is not harmful, and can
only speed up the convergence of gradient descent.

If θ0 and θ1 are initialized at
the global minimum, then one iteration will not change their values.

If the first few iterations of gradient descent cause f(θ0,θ1) to
increase rather than decrease, then the most likely cause is that we have set the
learning rate α to too large a value.
5.
Suppose that for some linear regression problem (say, predicting housing prices as in the lecture), we
have some training set, and for our training set we managed to find some θ0, θ1 such that J(θ0,θ1)=0. Which
of the statements below must then be true? (Check all that apply.)

This is not possible: By the definition of J(θ0,θ1), it is not possible for there to exist
θ0 and θ1 so that J(θ0,θ1)=0

For this to be true, we must have θ0=0 and θ1=0
so that hθ(x)=0

We can perfectly predict the value of y even for new examples that we have not yet seen.
(e.g., we can perfectly predict prices of even new houses that we have not yet seen.)

For these values of θ0 and θ1 that satisfy J(θ0,θ1)=0,
we have that hθ(x(i))=y(i) for every training example (x(i),y(i))