Question

1. The following are samples of a normal distribution with mean θ and variance σ, both are unknown:

54.1, 53.3, 55.9, 56.0, 55.7.

Find an unbiased estimate for θ.

2. With the data given in Problem 1, with θ unknown, find an unbiased estimate for σ

3. With the data given in Problem 1, with σ

2 unknown, find a 95% confidende interval for θ

Answer #1

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1. The...

Let Y be a normal random variable with mean μ and variance
σ2 . Assume that μ is known but σ2
is unknown. Show that ((Y-μ)/σ)2 is a pivotal
quantity. Use this pivotal quantity to derive a 1-α
confidence interval for σ2. (The answer should be left in
terms of critical values for the appropriate distribution.)

a.) Given a normal distribution with σ = 0.380. Find the
required sample size for a 95% confidence level (estimating the
mean), given a margin-of-error of 6%.
b.) Given the sample results taken from a normal population
distribution: mean = 4.65, σ = 0.32, and n = 17.
For a 99% confidence interval, find the margin-of-error for the
population mean. (use 2 decimal places)
c.) Given the sample results taken from a normal population
distribution: mean = 1.25, σ =...

Let X1,...,Xn be a random sample from a normal distribution with
mean zero and variance σ^2. Construct a 95% lower conﬁdence limit
for σ^2. Your anwser may be left in terms of quantiles of some
particular distribution.

Given the sample results taken from a normal population
distribution: mean = 4.65, σ = 0.32, and n = 13. Find the
margin-of-error and the 95% confidence interval for the population
mean. (use 2 decimal places)

Given estimator ?=cΣ(??−?̅)2 for ?2,
where ?2, represents variance of a normal
distribution whose mean and variance are both unknown.
a. Find c that gives the minimum-MSE estimator ?∗for
?2.
b. Is ?∗ MSE-consistent? Why or why not?

Let X1,...,Xn be a random sample from a normal
distribution where the variance is known and the mean is
unknown.
Find the minimum variance unbiased estimator of the
mean. Justify all your steps.

a. What is the standard error of a sampling
distribution? (out of the following)
the mean, the probability, the bias, the standard deviation, or
the variance
b. What is the standard deviation of a sampling
distribution called? (out of the following)
the spread, the variance, the standard error, the mean, the
standard variance
c. List two unbiased estimators and their
corresponding parameters. (Select all that apply out of the
following.)
μ is an unbiased estimator for x-bar, p is an...

A sample from a Normal distribution with an unknown mean µ and
known variance
σ = 45 was taken with n = 9 samples giving sample mean of ¯ y =
3.6.
(a) Construct a Hypothesis test with significance level α = 0.05 to
test whether the
mean is equal to 0 or it is greater than 0. What can you conclude
based on the
outcome of the sample?
(b) Calculate the power of this test if the true value...

Note given a Normal(θ, 1) distribution: 28, 33, 22, 35, 31
--> we want to estimate θ by minimizing residuals. Using the L2
norm squared;
1. What is the function sp(θ) we would like to minimize?
2. Graph sp(θ).
3. Using the Bisection Method find the Minimum Residual
Estimator for θ correct, 2 dec. places.
4. If using Newton’s Method for this optimization problem, what
is the refinement increment h(t)?

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