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To comply with food labelling regulations, a manufacturer of certain product (e.g. chocolate in packs) must prove that 99 % of its produced packs are each within 7g of the stated weight μ = 250g. The manufacturer knows that the average weight of its produced packs equals the stated weight. It also knows that the machinery is designed such that the respective weights of the products are normally distributed with mean equal to the stated weight and an unknown variance. To test the machinery is working to the specification, the company randomly selected n = 100 packs from the production lines and calculated the residual (y) weight. They reported the sum of squared residuals SSR = (i - μ)² was 572.78. a) Identify a parameter whose value will allow you to answer the question about the precision of manu- facturing you wish to make inference on? By appropriate manipulation of the likelihood, demonstrate that SSR is the sufficient statistic for 0. b) By choosing an appropriate one-to-one transformation f(0) of the parameter identified in a), write down a conjugate prior for the new parameter. Hint: The prior will be defined by two hyper-parameters. c) Determine the posterior distribution p(f(0)|y1,..., Yn). Substituting 1 for both hyper-parameters, determine the 95% central credible interval for 0. d) For the posterior distribution in b), determine the 95 % highest posterior density interval for 0, and compare it to the 95 % central credible interval. e) Do you believe, based on posterior inference, that the machinery used by this manufacturer satisfies the requirement that 99 % of its products with the stated weights 250g per pack are within 7g of the stated weight?

To comply with food labelling regulations, a manufacturer of certain product (e.g. chocolate in packs) must prove that 99 % of its produced packs are each within 7g of the stated weight μ = 250g. The manufacturer knows that the average weight of its produced packs equals the stated weight. It also knows that the machinery is designed such that the respective weights of the products are normally distributed with mean equal to the stated weight and an unknown variance. To test the machinery is working to the specification, the company randomly selected n = 100 packs from the production lines and calculated the residual (y) weight. They reported the sum of squared residuals SSR = (i - μ)² was 572.78. a) Identify a parameter whose value will allow you to answer the question about the precision of manu- facturing you wish to make inference on? By appropriate manipulation of the likelihood, demonstrate that SSR is the sufficient statistic for 0. b) By choosing an appropriate one-to-one transformation f(0) of the parameter identified in a), write down a conjugate prior for the new parameter. Hint: The prior will be defined by two hyper-parameters. c) Determine the posterior distribution p(f(0)|y1,..., Yn). Substituting 1 for both hyper-parameters, determine the 95% central credible interval for 0. d) For the posterior distribution in b), determine the 95 % highest posterior density interval for 0, and compare it to the 95 % central credible interval. e) Do you believe, based on posterior inference, that the machinery used by this manufacturer satisfies the requirement that 99 % of its products with the stated weights 250g per pack are within 7g of the stated weight?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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To comply with food labelling regulations, a manufacturer of certain product (e.g. chocolate in packs) must prove that 99 % of its produced packs are each within 7g of the stated weight μ = 250g. The manufacturer knows that the average weight of its produced packs equals the stated weight. It also knows that the machinery is designed such that the respective weights of the products are normally distributed with mean equal to the stated weight and an unknown variance. To test the machinery is working to the specification, the company randomly selected n = 100 packs from the production lines and calculated the residual (y) weight. They reported the sum of squared residuals SSR = (i - μ)² was 572.78. a) Identify a parameter whose value will allow you to answer the question about the precision of manu- facturing you wish to make inference on? By appropriate manipulation of the likelihood, demonstrate that SSR is the sufficient statistic for 0. b) By choosing an appropriate one-to-one transformation f(0) of the parameter identified in a), write down a conjugate prior for the new parameter. Hint: The prior will be defined by two hyper-parameters. c) Determine the posterior distribution p(f(0)|y1,..., Yn). Substituting 1 for both hyper-parameters, determine the 95% central credible interval for 0. d) For the posterior distribution in b), determine the 95 % highest posterior density interval for 0, and compare it to the 95 % central credible interval. e) Do you believe, based on posterior inference, that the machinery used by this manufacturer satisfies the requirement that 99 % of its products with the stated weights 250g per pack are within 7g of the stated weight?
Transcribed Image Text:To comply with food labelling regulations, a manufacturer of certain product (e.g. chocolate in packs) must prove that 99 % of its produced packs are each within 7g of the stated weight μ = 250g. The manufacturer knows that the average weight of its produced packs equals the stated weight. It also knows that the machinery is designed such that the respective weights of the products are normally distributed with mean equal to the stated weight and an unknown variance. To test the machinery is working to the specification, the company randomly selected n = 100 packs from the production lines and calculated the residual (y) weight. They reported the sum of squared residuals SSR = (i - μ)² was 572.78. a) Identify a parameter whose value will allow you to answer the question about the precision of manu- facturing you wish to make inference on? By appropriate manipulation of the likelihood, demonstrate that SSR is the sufficient statistic for 0. b) By choosing an appropriate one-to-one transformation f(0) of the parameter identified in a), write down a conjugate prior for the new parameter. Hint: The prior will be defined by two hyper-parameters. c) Determine the posterior distribution p(f(0)|y1,..., Yn). Substituting 1 for both hyper-parameters, determine the 95% central credible interval for 0. d) For the posterior distribution in b), determine the 95 % highest posterior density interval for 0, and compare it to the 95 % central credible interval. e) Do you believe, based on posterior inference, that the machinery used by this manufacturer satisfies the requirement that 99 % of its products with the stated weights 250g per pack are within 7g of the stated weight?
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