Reference no: EM13908737
1) Let X1, X2, ..... , Xn be a random sample from a population with true mean, µ and true variance, σ2. Let the sample mean be estimated as
X = a1X1 + a2X2 + ....... + anXn,
where a1, a2, cdots, an are non-negative constants.
a) What is the condition, a1, a2,....., an must satisfy so that X is an unbiased estimator of µ?
b) Find the variance of X.
c) What are the conditions, a1, a2,....., an must satisfy so that X is a minimum variance unbiased estimator of µ? Just give the condition. You do not have to solve and show when exactly the condition will be met. Your solution will be of the form, some expression is minimum subject to some constraint.
2) Let X1 be the sample mean of n samples of a normal population with mean, µ and variance σ2. Let X2 be the sample mean of n samples of a normal population with mean µ and variance, σ2. Let the two sets of samples be independent of each other.
a) Show that, ∀ ω ∈ [0, 1] (i.e., 0 ≤ ω ≤ 1), ωX'1 + (1 - ω)X'2 is an un-biassed estimator of µ.
b) Show that the variance of this estimator is minimum when
ω = σ22/σ12 + σ22
3) A random sample of the performance of 100 students in a test is taken. The true mean, µ was found to be 75 and the true variance, σ2, was found to be 256.
a) Find the probability that the sample mean, X' falls between 67 and 83.
b) Repeat part #3a) using Chebyshev inequality.
4) A telecommunication network design Engineer wishes to determine the channel capacity of a wireless channel. She performs n = 16 experiments and finds the sample average capacity (based on these 16 experiments) to be 14 Mbps.
a) If the true variance of the channel capacity is 16 (Mbps)2, then what would be the range of the true channel capacity if the Engineer wishes to be 95% confident.
b) if she used the sample variance (which is also 16 (Mbps)2), instead of the true variance.
5) Consider the following data of lengths (in feet) of steel beams rolled in a steel mill.
11.8, 12.1, 12.5, 11.7, 11.9, 12.0, 12.2, 11.5, 11.9, 12.2.
a) Find the error in the claim that the average length of the beam is 12 feet.
b) If you want to make a claim about the mean length of the manufactured steel beams with 99% confidence, what would be the range of the mean?
|
Display currency conversion for us dollar
: Write a MATLAB script to display currency conversion for US Dollar, British Pound, Euro with Singapore Dollar as the base Currency. Your output should show the equivalent values for USD, GBP, EURO for every Singapore dollar in increments of 1 doll..
|
|
Implement distributed system facility
: In this assignment you will implement a distributed system facility based on Client-Server programming - The client and server processes will be on different machines. In this case, the server is a stateful server.
|
|
Predict the level of the ocean
: Write a mathematical expression that would predict the level of the ocean at this beach at any time of the day. Find times in the day when the ocean level is exactly at sea level.
|
|
Is the prospect a positive npv using roe as the hurdle rate
: Using tax cash flows and no debt (pure equity), is prospect a positive NPV using ROE as the hurdle rate? Using the after tax cash flows and the firm's WACC, is this project desirable? Explain how you came to this conclusion.
|
|
What would be the range of the mean
: If you want to make a claim about the mean length of the manufactured steel beams with 99% confidence, what would be the range of the mean - Find the error in the claim that the average length of the beam is 12 feet.
|
|
Question regarding the intervalsum
: Write a function named intervalSum. This function must not accept arguments but it will ask the user to input two integers m and n and calculate and display the sum of all the integers from min(m, n) to max(m, n)
|
|
What happens to the present value of an annuity
: As you increase the length of the time involved, what happens to the present value of an annuity? What happens to the future value? Tri-State Megabucks Lottery advertises a $10 million grand prizes.
|
|
Calculate the test statistic
: An investigator analyzed the leading digits of the amounts from 200 checks issued by three suspect companies. Calculate the χ2 test statistic. Calculate the χ2critical value
|
|
What are the intercepts of the line
: What are the intercepts of the line - what is the slope of the line and write a linear equation for the line.
|