Question

For 113 consecutive days, a process engineer has measured the temperature of champagne bottles as they are made ready for serving. Each day, she took a sample of 9 bottles. The average across all 1017 bottles (113 days, 9 bottles per day) was 60 degrees Fahrenheit. The standard deviation across all bottles was 0.5 degrees. (Round your intermediate calculations and final answer to 4 decimal places.) When constructing an X-bar chart, what would be the upper control limit

Answer 1

Answer:

upper control limit $UCL=7.1632$

Concept :

$UCL=\bar{\bar{x}}+A_2\bar{R}$  where $A_2=0.34$

$\bar{\bar{x}}=\frac{\bar{x}}{n}$

$S=\frac{\bar{R}}{d}\Rightarrow \bar{R}=S\times d$ where $S=0.5, d=2.97$

Given :

She took a sample of per day $(n)$ is $9$ bottles.

The standard deviation $S$of all bottles is $0.5$ degree.

Average across all $1017$ bottles $\bar{x}$are $60$ degree Fahrenheit.

To find :

The value of the upper control limit (UCL)

Explanation :

Here, firstly we find the value of $\bar{\bar{x}}$ .

$\because\bar{\bar{x}}=\frac{\bar{x}}{n}$

$\therefore \bar{\bar{x}}=\frac{60}{9}$

$\Rightarrow \bar{\bar{x}}=6.66$ and $\bar{R}=S\times d$

$\Rightarrow \bar{R}=0.5\times2.97=1.48$

$\Rightarrow \bar{R}=1.48$

Now, $UCL=\bar{\bar{x}}+A_2\bar{R}$

$\Rightarrow UCL=6.66+0.34\times1.48$

$\Rightarrow UCL=6.66+0.5032$

$\Rightarrow UCL=7.1632$ degrees.