Homework 1: Prerequisite

Goals

Homework 1 is meant to give you a concrete understanding of the prerequisites for this class. If you find these challenging, you should reach out to the TA to brush up on these topics.

Q1 - Loading data and applying functions

Please calculate the following statistics for the noise complaints from the dataset graffiti_vs_noise_by_zip_2018.csv on Canvas. You do not need to submit your code but please round the answer to the 5th decimal.

• The mean
• The sample standard deviation (with the n-1 adjustment)
• The mean for zip codes between 11000 and 11500 (inclusive, i.e. including 11000)
• The sample standard deviation for zip codes between 11000 and 11500

Q2 - Reviewing definitions

Please express \(Var(X + Y)\) using \(Var(X)\), \(Var(Y)\), and \(Cov(X, Y)\).

Q3 - Probability review

Imagine I have a coin with \(P(coin toss = Heads) = p\), write down the probability that I have k heads after tossing the coin n times.

Q4 - Calculus review in finding maximums

Continuing Q3, what value of p would maximize the probability above? Please express the answer in terms of k and n.

Q5 - Logic review

Small reminder, 2 distributions can be the same type of distribution but may not be the same distribution if they do not share the same set of parameters.

• True/False, if the means are the same between two distributions, they must be the same distribution. No proof necessary.
• True/False, if the means are different between two distributions, they must be different distributions. No proof necessary.

Q6 - Algebra with random variables

Let \(Y = f(X)\) where \(X\) is a random variable with \(SD(X)>0\), please mathematically write down a solution for \(f()\) such that \(E(Y)=0\) and \(SD(Y)=1\).

Q7 - Simulation and hypothesis testing review

• Please write the code that would simulate 100 data points from a random variable \(X\sim Normal(0, 1^2)\) and another 50 data points from a random variable \(Y\sum Normal(0.1, 1.1^2)\).
• Pretending we do not know the parameters used to generate the data, please construct the 87% confidence interval for the expectation of X using the data you simulated. Please show your code.
• Use at most 2 sentences to explain why we should “pretend we do not know the parameters used to generate the data” when constructing the confidence interval?
• Pretending that we do not know the parameters used to generate the data, we want to test the idea whether the expectation of X is the same as the expectation of Y.
  • What is your null hypothesis?
  • Please write the code that would calculate the p-value from a 2 sample t-test that does not assume the variances are identical. Remember to report your p-value.