Question

Question 4 (6 marks) Suppose the compressive strength of batch of concrete is normally distributed with an unknown mean and known standard deviation of 4 MPa. Suppose 20 compressive strength tests are conducted and the arithmetic (or sample) mean is found to be 37.5 MPa. Plot the prior and posterior density functions for the mean strength of concrete for the following three priors: 1) Prior mean is 35 MPa and prior standard deviation is 5 MPa. 2) Prior mean is 35 MPa and prior standard deviation is 0.5 MPa. 3) Prior mean is 35 MPa and prior standard deviation is 0.1 MPa. Compare the results.

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Answer 1

"Given Region that: - Let be the compresherive strength is written b[x] Here x similar to N [mue_1 sigma^2] where sigma = 4 prior of mue similar to N [mue_0 , sigma^2_0] Here Absolute that 20 samples are tested sample Mean is 37.5 Let x-1 ......x_n similar to N [mue sigma^2] mue similar to N [mue_0 sigma^2_0] L [mue /x sigma^2] = 1/[Squareroot of 2x)^n sigma^n e -1/2 sigma r prior that pie (H 0 = 1/squareroot 2x pie_0 e^-1/2 [mue - mue_0]^2/sigma_0^2 Therefore L[mue/x, sigma^2] x [pie (mue)] where 1 /(squareroot 2 pie)^n sigma^n . 1/squareroot 2x sigma exp -1/2 sigma^2 sigma 6-1_1 < 1 [x^2-I - 2.u xi + mue^2] = -1/2 sigma^2_0 [mue^2 - 2 mue_0 + mue^2_0] \r\n exp [-1/2 [sigma^1_