Hi, so I've been working on a question involving constructing a mean and range chart with upper and lower action and warning limits, I've got the first part of the question right (I've plotted the chart with the correct limits) but I can't seem to make any progress on the last part involving Cp and Cpk values.

Process mean x̅ = 50.06
Mean Range = 1.77

Standard Deviation SD = 0.8596406
Standard Error = 0.4298203

Mean Chart
Upper Action Limit= 51.3494609
Upper Warning Limit= 50.9196406
Lower Warning Limit= 49.2003594
Lower Action Limit= 48.7705391
Range Chart
UAL= 4.5489
UWL= 3.4161
LWL= 0.5133
LAL= 0.177

The values above should all be correct, they match the answers on the markscheme

The first question is: with a specified mean of 49, what is the minimum tolerance (T)? and confirm this by calculating Cpk.

I have the answer from the markscheme however it doesn't give a method (the answer should be T = 3.63892181)

The second part is: with a specified tolerance of +/- 4 what is the maximum and minimum values for the specified mean? Confirm the answer by calculating Cpk.

Again I have the given answers see below:

Maximum specified mean = 51.4810782 Cpk min ( 2.10207156 , 1 )

Minimum specified mean = 48.6389218 Cpk min ( 1 , 2.10207156 )

but no method so I'm struggling to figure out how to get to the answers

These are the equations I have but I've had no luck in getting to the answer with them:

Standard deviation = mean range / hartley's constant (in this case hartley's = 2.059)
Standard error = standard deviation / √n ( sample size n = 4)

Cp = 2T / 6 x standard deviation ( T is tolerance)

upper standard limit USL= specified mean + T
lower standard limit LSL= specified mean - T

Cpk = min ( (USL - x̅) / 3 x SD , (x̅ - LSL) / 3 x SD )

(all x symbols are multiply signs in the equations I've listed)

I don't know if I'm missing an equation or anything but any help you guys can give me would be fantastic,