Calculate the mean (x) for each sample group by summing the lengths and dividing by the number of studs (5).

You're on a roll — here's the solution: b) (i) Step 1: Calculate the mean (x) for each sample group by summing the lengths and dividing by the number of studs (5). Step 2: Calculate the range (R) for each sample group by subtracting the minimum length from the maximum length. For Sample Group A: x_A = (28+27+29+27+28)/(5) = (139)/(5) = 27.8 mm R_A = 29 - 27 = 2 mm For Sample Group B: x_B = (27+29+28+26+28)/(5) = (138)/(5) = 27.6 mm R_B = 29 - 26 = 3 mm For Sample Group C: x_C = (29+27+26+28+29)/(5) = (139)/(5) = 27.8 mm R_C = 29 - 26 = 3 mm For Sample Group D: x_D = (28+26+29+26+29)/(5) = (138)/(5) = 27.6 mm R_D = 29 - 26 = 3 mm For Sample Group E: x_E = (28+27+26+28+27)/(5) = (136)/(5) = 27.2 mm R_E = 28 - 26 = 2 mm b) (ii) Step 1: Calculate the process mean (x) by averaging the means of all sample groups. Step 2: Calculate the mean of sample ranges (R) by averaging the ranges of all sample groups. Process mean: x = x_A + x_B + x_C + x_D + x_E5 = (27.8 + 27.6 + 27.8 + 27.6 + 27.2)/(5) = (138)/(5) = 27.6 mm Mean of sample ranges: R = (R_A + R_B + R_C + R_D + R_E)/(5) = (2 + 3 + 3 + 3 + 2)/(5) = (13)/(5) = 2.6 mm b) (iii) Step 1: Calculate the process standard deviation () using the formula = Rd_n. Step 2: Calculate the standard error of the mean, ()/(sqrt(n)), where n=5 is the sample size. Step 3: Determine the warning and action lines for the mean chart using the formulas: Center Line (CL) = x Upper Warning Line (UWL) = x + 2 ()/(sqrt(n)) Lower Warning Line (LWL) = x - 2 ()/(sqrt(n)) Upper Action Line (UAL) = x + 3 ()/(sqrt(n)) Lower Action Line (LAL) = x - 3 ()/(sqrt(n)) Given Hartley's constant, d_n = 2.33. Process standard deviation: = Rd_n = (2.6)/(2.33) ≈ 1.11588 mm Standard error of the mean: ()/(sqrt(n)) = (1.11588)/(sqrt(5)) ≈ (1.11588)/(2.23607) ≈ 0.49903 mm Center Line (CL): CL = x = 27.6 mm Warning Lines: UWL = 27.6 + 2(0.49903) = 27.6 + 0.99806 = 28.598 mm LWL =