Question

For the question where should the control limits in an x-bar chart be placed if the design process sets α = 0.01 with the following parameters. average is 100, standard deviation is 20 and n = 25? I need to find z_α/2 . the book says α = 0.01 puts the control limits at z_{α​​​​​​​/2} = (+/-)2 standard errors from the process target. where does the (+/-) 2 come from?

Answer 1

The book says to puts the control limits at z_α/2 = (+/-)2 standard errors. Then α should be 0.05

α = 0.05

α/2 = 0.05/2 = 0.025

From z table, z_α/2 = 1.96 $\approx$ 2

Margin of error = z_α/2 * standard errors = 2 * standard errors

So, the control limits are set at -2 * standard errors and +2 * standard errors

Lower Control limit = x-bar - 2 * standard error

Upper Control limit = x-bar + 2 * standard error

Now for α = 0.01

α/2 = 0.01/2 = 0.005

From z table, z_α/2 = 2.576

Margin of error = z_α/2 * standard errors = 2.576 * standard errors

standard error =
ulo
= 20 / $\sqrt{25}$ = 4

Lower Control limit = x-bar - 2.576 * standard error = 100 - 2.576 * 4 = 89.696

Upper Control limit = x-bar + 2.576 * standard error = 100 + 2.576 * 4 =  110.304