409Pr1 - 1. Let X 1 , X 2 , , X n be a random sample from...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
1. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( 29 4 3 θ 4 θ x e x x f - = x > 0 θ > 0. a) Find the sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for θ . b) Obtain the maximum likelihood estimator of θ , θ ˆ . c) Is θ ˆ a consistent estimator of θ ? Justify your answer . d) Is θ ˆ an unbiased estimator of θ ? Justify your answer . e)* What is the probability distribution of Y = = n i i 1 4 X ? f)* Suggest a ( 1 - α ) 100 % confidence interval for θ based on Y = = n i i 1 4 X . 2. Let λ > 0 be an unknown parameter and let X 1 , X 2 , … , X n be independent random variables, each with the probability density function f ( x ) = ( 29 < < - - otherwise 0 1 0 1 1 λ λ x x . a) Find the sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for λ . b) Obtain the maximum likelihood estimator of λ , n λ ˆ . c) Is n λ ˆ is a consistent estimator for λ ? d) Obtain the method of moments estimator of λ , n λ ~ . e) Is n λ ~ is a consistent estimator for λ ?
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
3. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability mass function P ( X i = 1 ) = θ θ 3 + , P ( X i = 2 ) = θ 3 2 + , P ( X i = 3 ) = θ 3 1 + , θ > 0. a) Find a sufficient statistic for θ . b) Obtain the method of moments estimator θ ~ of θ . c) Obtain the maximum likelihood estimator θ ˆ of θ . 4. A store sells "16-ounce" boxes of Captain Crisp cereal. A random sample of 9 boxes was taken and weighed. The results were the following (in ounces): 15.5 16.2 16.1 15.8 15.6 16.0 15.8 15.9 16.2 Assume the weight of cereal in a box is normally distributed. Hint: Σ x = 143.1, Σ x 2 = 2275.79, Σ ( x x ) 2 = 0.50. a) Compute the sample mean x and the sample standard deviation s . b) Construct a 95% confidence interval for the overall average weight of boxes of Captain Crisp cereal. c) Construct a 95% confidence upper bound for the overall average weight of boxes of Captain Crisp cereal. d) Construct a 90% confidence interval for the overall standard deviation of the weights of boxes of Captain Crisp cereal. e) Construct a 99% confidence lower bound for the overall standard deviation of the weights of boxes of Captain Crisp cereal. 5. Starting annual salaries for college graduates of certain majors are believed to have a standard deviation of approximately $1800. What is the minimum required sample size to estimate the average annual salary for college graduates to within $300 with 90% confidence?
Background image of page 2
6. A researcher wishes to determine whether the starting salaries of high-school math teachers in private schools are different from those of high-school math teachers in public schools. She selects a sample of new math teachers from each type of school and calculates the sample means and sample standard deviations of their salaries. Assume that the populations are normally distributed and the population variances
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

Page1 / 14

409Pr1 - 1. Let X 1 , X 2 , , X n be a random sample from...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online