The thickness, SSS mm, of a mechanical shim has a normal distribution S∼N(1.45,0.042)S \sim N(1.45, 0.04^2)S∼N(1.45,0.042) and the thickness, KKK mm, of a spacer has a normal distribution K∼N(1.52,0.052)K \sim N(1.52, 0.05^2)K∼N(1.52,0.052). An engineer uses 5 independent shims and one spacer for a high-precision assembly. Find the probability that the combined thickness of the 5 shims is less than 5 times the thickness of the spacer.
Two independent random samples X1,X2,X3,X4,X5X_1, X_2, X_3, X_4, X_5X1,X2,X3,X4,X5 and Y1,Y2,Y3,Y4,Y5Y_1, Y_2, Y_3, Y_4, Y_5Y1,Y2,Y3,Y4,Y5 are each taken from a normal population with mean μ\muμ and standard deviation σ\sigmaσ.
Find the distribution of the random variable D=Y1−XˉD = Y_1 - \bar{X}D=Y1−Xˉ.
Hence show that P(Y1>Xˉ+σ)=0.1807P(Y_1 > \bar{X} + \sigma) = 0.1807P(Y1>Xˉ+σ)=0.1807 correct to 4 decimal places.
A researcher believes that P(U1>Uˉ+σ)=0.1807P(U_1 > \bar{U} + \sigma) = 0.1807P(U1>Uˉ+σ)=0.1807 for any random sample U1,U2,U3,U4,U5U_1, U_2, U_3, U_4, U_5U1,U2,U3,U4,U5 taken from the same normal population. Explain briefly why the result from part (b) should not be used to confirm the researcher's belief.
Find, correct to 3 decimal places, the actual value of P(U1>Uˉ+σ)P(U_1 > \bar{U} + \sigma)P(U1>Uˉ+σ).
