Question

In: Statistics and Probability

Suppose we have a random sample of size 2𝑛 from a population denoted by 𝑋, and 𝐸(𝑋) = 𝜇 and 𝑉(𝑋) = 𝜎^2 . Let

Suppose we have a random sample of size 2𝑛 from a population denoted by 𝑋, and 𝐸(𝑋) = 𝜇 and 𝑉(𝑋) = 𝜎² . Let

be two estimators of 𝜇. Which is the better estimator of 𝜇? Explain your choice.

Expert Solution

Solution

Suppose we have a random sample of size 2𝑛 from a population denoted by 𝑋, and 𝐸(𝑋) = 𝜇 and 𝑉(𝑋) = 𝜎² .

be two estimators of 𝜇. Which is the better estimator of 𝜇? Explain your choice

Then we have

Then follow by

These two estimators are unbiased. and we have some quation

These two estimators are unbiased.

Therefore:

SUN BUNRA answered 3 years ago

Let ?1,?2, … , ?? be a random sample of size ? from a population that...

Let ?1,?2, … , ?? be a random sample of size ? from a population that can be modeled by the following probability model: ??(?) = ?? ?−1 ? ? , 0 < ? < ?, ? > 0 a) Find the probability density function of ?(?) = max(?1,?2, … , ??). b) Is ?(?) an unbiased estimator for ?? If not, suggest a function of ?(?) that is an unbiased estimator for ?.

Consider an i.i.d. random sample of size 3 denoted by ?1,?2, ?3 from the same population,...

Consider an i.i.d. random sample of size 3 denoted by ?1,?2, ?3 from the same population, where the mean ? and variance ? 2 are unknown. Suppose that you have the following two different estimators for mean ?. (Remember: no work, no credit.) ?̂1 = 0.3?1 + 0.5?2 + 0.2?3 ?̂2 = 0.5?1 + 0.5?3 a. Is ?̂1 unbiased? b. Is ?̂2 unbiased? c. Which one is preferred, ?̂1 or ?̂2?

Suppose that a random sample of size 64 is to be selected from a population with...

Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5. (a) What are the mean and standard deviation of the x sampling distribution? μx = 1 40 Correct: Your answer is correct. σx = 2 .625 Correct: Your answer is correct. (b) What is the approximate probability that x will be within 0.2 of the population mean μ? (Round your answer to four decimal places.) P = 3...

Suppose that a random sample of size 64 is to be selected from a population with...

Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5. a) What is the approximate probability that x will differ from μ by more than 0.8? (Round your answer to four decimal places.)

Suppose that you have a random sample of size n from a population with Gamma density...

Suppose that you have a random sample of size n from a population with Gamma density with α= 3 but unknown β. Write down the likelihood function, and find a sufficient statistic. Find the MLE and the MOM estimators forβ. (Hint: They should be equal.) Then find the MSE for this estimator by finding the bias and the variance. Is it consistent? Is it MVUE? Explain why or why not.

Suppose, for a random sample selected from a normal population, we have the values of the...

Suppose, for a random sample selected from a normal population, we have the values of the sample mean x ̄ = 67.95 and the standard deviation s = 9. a. Construct a 95% confidence interval for population mean μ assuming the sample size n = 16. b. Construct a 90% confidence interval for population mean μ assuming n = 16. c. Obtain the width of the confidence intervals calculated in a and b. Is the width of 90% confidence interval...

1. Suppose that a random sample of size 64 is to be selected from a population...

1. Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5. a. What is the mean of the ¯xx¯ sampling distribution? b. What is the standard deviation of the ¯xx¯ sampling distribution? c. What is the approximate probability that ¯xx¯ will be within 0.5 of the population mean μμ? d. What is the approximate probability that ¯xx¯ will differ from μμ by more than 0.7? 2. A Food...

Suppose a simple random sample of size nequals200 is obtained from a population whose size is...

Suppose a simple random sample of size nequals200 is obtained from a population whose size is Upper N equals 15 comma 000 and whose population proportion with a specified characteristic is p equals 0.6 . (a) Describe the sampling distribution of ModifyingAbove p with caret. Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less...

1. A random sample of size n = 10 is taken from a large population. Let...

1. A random sample of size n = 10 is taken from a large population. Let μ be the unknown population mean. A test is planned of H0: μ=12vs. HA: μ̸=12usingα=0.1. A QQ plot indicates it is reasonable to assume a normal population. From the sample, x̄ = 14.2 and s = 4.88. (I suggest doing this problem with a calculator and table as practice for exams. You may check your answers with R if you wish.) (a) Since the...

Suppose a simple random sample of size n=75 is obtained from a population whose size is...

Suppose a simple random sample of size n=75 is obtained from a population whose size is N= 30,000 and whose population proportion with a specified characteristic is p= 0.4 . A) Determine the standard deviation of the sampling distribution of p hat (Round to 6 decimals) B) What is the probability of obtaining x=33 or more individuals with the characteristic? That is, what is P(p ≥0.44)? (Round to 4 decimals)

Question

Suppose we have a random sample of size 2𝑛 from a population denoted by 𝑋, and 𝐸(𝑋) = 𝜇 and 𝑉(𝑋) = 𝜎^2 . Let

Solutions

Expert Solution

Related Solutions

Let ?1,?2, … , ?? be a random sample of size ? from a population that...

Consider an i.i.d. random sample of size 3 denoted by ?1,?2, ?3 from the same population,...

Suppose that a random sample of size 64 is to be selected from a population with...

Suppose that a random sample of size 64 is to be selected from a population with...

Suppose that you have a random sample of size n from a population with Gamma density...

Suppose, for a random sample selected from a normal population, we have the values of the...

1. Suppose that a random sample of size 64 is to be selected from a population...

Suppose a simple random sample of size nequals200 is obtained from a population whose size is...

1. A random sample of size n = 10 is taken from a large population. Let...

Suppose a simple random sample of size n=75 is obtained from a population whose size is...