stats assignment 113, Assignments of Economics

Download stats assignment 113 and more Assignments Economics in PDF only on Docsity! ECO 5400: Statistics Instructor: Bipasha Maity Ashoka University Monsoon 2020 Homework Due: Monday, November 23, 2020 by 6 pm via email. Question 1: Use the table/simulation to find the following critical values: 𝑇~𝑡 Distribution: 1 − 𝑃(|𝑇| ≤ 𝑡𝑑𝑓) = 0.05 for 𝑑𝑓 = 5, 10, 20, 30, 50. How do these compare to the 0.05 critical values of the Normal distribution? Question 2: Suppose 𝑋1, 𝑋2, … . , 𝑋6 are six random variables that form a random sample from the standard normal distribution. Let 𝑌 = (𝑋1 + 𝑋2 + 𝑋3) 2 + (𝑋4 + 𝑋5 + 𝑋6) 2 Find the value of 𝑐 such that the random variable 𝑐𝑌 will follow the chi-square distribution Question 3: Suppose you are looking at treatment and control group differences to determine the effectiveness of a computer-skills training programme on weekly wages. The standard deviation of the weekly wages is $15 for both the treatment and control groups. Now the average weekly wages for the control group is $100 and $100+δ for the treatment group. If you are constrained to have 25% of your observations in the control group and 75% observations in your treatment group, how large does 𝑁 have to be in order for the probability of ?̅?𝑇 − ?̅?𝐶 > 0 is atleast 95%? Note: ?̅?𝑇 and ?̅?𝐶 are the sample averages weekly wages of the treatment and control groups respectively and 𝑁 is the total number of observations in your experiment that is split between the treatment and control groups. Also assume that the sample averages are normally distributed. Question 4: A random sample of 𝑛 items is to be taken from a distribution with mean µ and standard deviation 𝜎. a) Use the Chebyshev inequality to find out the smallest number of items 𝑛 that must be taken to satisfy: 𝑃(|?̅?𝑛 − µ| ≤ 𝜎 4 ) ≥ 0.99 b) Use the central limit theorem to find out the smallest number of items 𝑛 that must be taken to satisfy the above relation in a) approximately. Question 5: MLE and MoM estimators: a) Give the log likelihood function of a sample of 𝑁 iid Poisson random variables 𝑋1, … . , 𝑋𝑁 and solve for the MLE estimator of the parameter λ. b) Provide the MoM estimator of the parameter λ and compare with the MLE estimator. Question 6: a) Suppose that the ransom variables 𝑋1, 𝑋2, … . , 𝑋𝑛 form a random sample from the Bernoulli distribution with parameter θ, which is unknown (0 ≤ θ ≤ 1). For all observed values 𝑥1, 𝑥2, … . , 𝑥𝑛 where each 𝑥𝑖 is either 0 or 1, what is the likelihood function? What is the MLE of θ? b) It is not known what proportion 𝑝 of the purchases of a certain brand of breakfast cereal are made by women and what proportion is made by men. In a random sample of 70 purchases of this cereal, it was found that 58 were made by women and the rest by men. Find the MLE of 𝑝.