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stats assignment 113, Assignments of Economics

Acharya N. G. Ranga Agricultural UniversityEconomics

Statistics is a branch of applied mathematics dealing with data collection, organization, analysis, interpretation and presentation. Descriptive statistics summarize data. ... In addition to being the name of a field of study, the word "statistics" also refers to numbers that are used to describe data or relationships.

Typology: Assignments

2020/2021

Uploaded on 02/26/2021

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vrinda-juneja 🇮🇳

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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 𝑋
𝑇− 𝑋
𝐶> 0 is atleast
95%?
Note: 𝑋
𝑇 and 𝑋
𝐶 are the sample averages weekly wages of the treatment and
control groups respectively and 𝑁 is the total number of observations in your
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ECO 5 4 00: Statistics

Instructor: Bipasha Maity

Ashoka University

Monsoon 20 20

Homework Due: Monday, November 23, 20 20 by 6 pm via email.

Question 1 : Use the table/simulation to find the following critical values:

𝑇~𝑡 Distribution: 1 − 𝑃(

𝑑𝑓

) = 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 𝑋

1

2

6

are six random variables that form a random

sample from the standard normal distribution.

Let 𝑌 = (𝑋

1

2

3

2

4

5

6

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 $

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 𝑋

𝑇

𝐶

0 is atleast

Note: 𝑋

𝑇

and 𝑋

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:

𝑛

− μ| ≤

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 𝑋

1

𝑁

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 𝑋

1

2

𝑛

form a random

sample from the Bernoulli distribution with parameter θ, which is unknown ( 0 ≤

θ ≤ 1 ). For all observed values 𝑥 1

2

𝑛

where each 𝑥

𝑖

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.