# ▸ Linear Regression with Multiple Variables :

1. Suppose m=4 students have taken some classes, and the class had a midterm exam and a final exam. You have collected a dataset of their scores on the two exams, which is as follows:

You’d like to use polynomial regression to predict a student’s final exam score from their midterm exam score. Concretely, suppose you want to fit a model of the form $\inline&space;h_{\theta}(x)&space;=&space;\theta_{0}&space;+&space;\theta_{1}&space;x_{1}&space;+&space;\theta_{2}&space;x_{2}$, where $\inline&space;x_1$ is the midterm score and x_2 is (midterm score)^2. Further, you plan to use both feature scaling (dividing by the “max-min”, or range, of a feature) and mean normalization.
What is the normalized feature $\inline&space;x_2^{(4)}$ ? (Hint: midterm = 69, final = 78 is training example 4.) Please round off your answer to two decimal places and enter in the text box below.
2. -0.47

1. You run gradient descent for 15 iterations with $\inline&space;\alpha&space;=&space;0.3$ and compute after each iteration. You find that the value of $\inline&space;J(\theta)$ decreases slowly and is still decreasing after 15 iterations. Based on this, which of the following conclusions seems most plausible?

1. You run gradient descent for 15 iterations with $\inline&space;\alpha&space;=&space;0.3$ and compute after each iteration. You find that the value of $\inline&space;J(\theta)$ decreases quickly then levels off. Based on this, which of the following conclusions seems most plausible?

1. Suppose you have m = 23 training examples with n = 5 features (excluding the additional all-ones feature for the intercept term, which you should add). The normal equation is $\inline&space;\theta&space;=&space;(X^{T}&space;X)^{-1}X^{T}y$. For the given values of m and n, what are the dimensions of $\inline&space;\theta$, X, and y in this equation?
• X is 23 × 5, y is 23 × 1, θ is 5 × 1

1. Suppose you have a dataset with m = 1000000 examples and n = 200000 features for each example. You want to use multivariate linear regression to fit the parameters $\inline&space;\theta$ to our data. Should you prefer gradient descent or the normal equation?
• Gradient descent, since it will always converge to the optimal θ.
• Gradient descent, since $\inline&space;(X^T&space;X)^{-1}$ will be very slow to compute in the normal equation.
• With n = 200000 features, you will have to invert a 200001 x 200001 matrix to compute the normal equation. Inverting such a large matrix is computationally expensive, so gradient descent is a good choice.
• The normal equation, since it provides an efficient way to directly find the solution.
• The normal equation, since gradient descent might be unable to find the optimal θ.

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1. Which of the following are reasons for using feature scaling?
• It speeds up solving for θ using the normal equation.
• The magnitute of the feature values are nsignificant in terms of computational cost.
• It speeds up gradient descent by making it require fewer iterations to get to a good solution.
• Feature scaling speeds up gradient descent by avoiding many extra iterations that are required when one or more features takes on much larger values than he rest.

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