Answer to Question #152667 in Statistics and Probability for azizah

Question

A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that the process standard deviation is 1.5 ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appear below.

Weight
Day Package 1 Package 2 Package 3 Package 4
Monday 23 22 23 20
Tuesday 23 21 19 21
Wednesday 20 19 20 21
Thursday 18 19 20 17
Friday 18 20 22 22

(a) Calculate all sample means and the mean of all sample means.

(b) Calculate upper and lower 2-sigma x-bar chart control limits that allow for natural variations.
(c) Based on the x-bar chart, is this process in control? Explain either natural variation causes or assignable variation causes.

Answer 1

(a) Let's calculate sample means:

$\overline{x}_{Mon} = (23+22+23+20)/4=22$

$\overline{x}_{Tue} = (23+21+19+21)/4=21$

$\overline{x}_{Wed} = (20+19+20+21)/4=20$

$\overline{x}_{Thu} = (18+19+20+17)/4=18.5$

$\overline{x}_{Fri} = (18+20+22+22)/4=20.5$

Mean of all sample means:

$\overline{\overline{x}} = (22+21+20+18.5+20.5)/5=20.4$

(b) Upper and lower 2-sigma x-bar chart control limits:

$UCL=\overline{\overline{x}}+2*\frac{\sigma}{\sqrt{n}}=20.4+2*\frac{1.5}{\sqrt{4}}=21.9$

$LCL=\overline{\overline{x}}-2*\frac{\sigma}{\sqrt{n}}=20.4-2*\frac{1.5}{\sqrt{4}}=18.9$

(c) $\overline{x}_{Mon}$ is above UCL(X-bar) and $\overline{x}_{Thu}$ is below LCL(X-bar). The process is out of control. So, we have assignable variation causes.