Answer to Question #318014 in Statistics and Probability for rainyyyyy

Question

In building an arena, steel bars with a mean ultimate tensile strength of 400 megapascal (MPa) with a variance of 81 MPa were delivered by the manufacturer. The project engineer tested 50 steel bars and found out that the mean ultimate strenght is 390 MPa. The decision for the extension of the contract with the manufacturer depends on the engineer. Test the hypothesis whether there is no significant difference between the two means using a two - tailed with a = 0.01

Answer 1

$H_0:\mu =400\\H_1:\mu \ne 400\\Z=\sqrt{n}\frac{\bar{x}-\mu}{\sigma}=\sqrt{50}\frac{390-400}{\sqrt{81}}=-7.85674\sim N\left( 0,1 \right) \\P-value:\\P\left( \left| Z \right|>7.85674 \right) =2\varPhi \left( -7.85674 \right) =2\cdot 1.9713\cdot 10^{-15}<0.01\Rightarrow \\\Rightarrow \mu \ne 400$