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

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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 ?

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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?
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 are equal. Construct a 95% confidence interval for the difference in average starting salaries of high-school math teachers in private and public schools.

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

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