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

here's the solution:

b) (i)

Step 1: Calculate the mean ($\bar{x}$) 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:
$\bar{x}_A = \frac{28+27+29+27+28}{5} = \frac{139}{5} = 27.8 \text{ mm}$
$R_A = 29 - 27 = 2 \text{ mm}$

For Sample Group B:
$\bar{x}_B = \frac{27+29+28+26+28}{5} = \frac{138}{5} = 27.6 \text{ mm}$
$R_B = 29 - 26 = 3 \text{ mm}$

For Sample Group C:
$\bar{x}_C = \frac{29+27+26+28+29}{5} = \frac{139}{5} = 27.8 \text{ mm}$
$R_C = 29 - 26 = 3 \text{ mm}$

For Sample Group D:
$\bar{x}_D = \frac{28+26+29+26+29}{5} = \frac{138}{5} = 27.6 \text{ mm}$
$R_D = 29 - 26 = 3 \text{ mm}$

For Sample Group E:
$\bar{x}_E = \frac{28+27+26+28+27}{5} = \frac{136}{5} = 27.2 \text{ mm}$
$R_E = 28 - 26 = 2 \text{ mm}$

b) (ii)

Step 1: Calculate the process mean ($\bar{\bar{x}}$) by averaging the means of all sample groups.

Step 2: Calculate the mean of sample ranges ($\bar{R}$) by averaging the ranges of all sample groups.

Process mean:

$$\bar{\bar{x}} = \frac{\bar{x}_A + \bar{x}_B + \bar{x}_C + \bar{x}_D + \bar{x}_E}{5} = \frac{27.8 + 27.6 + 27.8 + 27.6 + 27.2}{5} = \frac{138}{5} = \boxed{27.6 \text{ mm}}$$

Mean of sample ranges:

$$\bar{R} = \frac{R_A + R_B + R_C + R_D + R_E}{5} = \frac{2 + 3 + 3 + 3 + 2}{5} = \frac{13}{5} = \boxed{2.6 \text{ mm}}$$

b) (iii)

Step 1: Calculate the process standard deviation ($\sigma$) using the formula $\sigma = \frac{\bar{R}}{d_n}$.

Step 2: Calculate the standard error of the mean, $\frac{\sigma}{\sqrt{n}}$, 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) = $\bar{\bar{x}}$
Upper Warning Line (UWL) = $\bar{\bar{x}} + 2 \frac{\sigma}{\sqrt{n}}$
Lower Warning Line (LWL) = $\bar{\bar{x}} - 2 \frac{\sigma}{\sqrt{n}}$
Upper Action Line (UAL) = $\bar{\bar{x}} + 3 \frac{\sigma}{\sqrt{n}}$
Lower Action Line (LAL) = $\bar{\bar{x}} - 3 \frac{\sigma}{\sqrt{n}}$

Given Hartley's constant, $d_n = 2.33$.

Process standard deviation:

$$\sigma = \frac{\bar{R}}{d_n} = \frac{2.6}{2.33} \approx 1.11588 \text{ mm}$$

Standard error of the mean:

$$\frac{\sigma}{\sqrt{n}} = \frac{1.11588}{\sqrt{5}} \approx \frac{1.11588}{2.23607} \approx 0.49903 \text{ mm}$$

Center Line (CL):

$$CL = \bar{\bar{x}} = \boxed{27.6 \text{ mm}}$$

Warning Lines:

$$UWL = 27.6 + 2(0.49903) = 27.6 + 0.99806 = \boxed{28.598 \text{ mm}}$$ $$$$