Question

For a bank processing new accounts with an intended average time of 10 minutes each, if five samples of seven observations (n) each have been taken, how do you construct the upper and lower control limits for both the mean (Xˉ)?

Answer 1

The upper control limit (UCL) for the mean
`(\(\bar{X}\))` is
`\( \bar{X} + A_2 * (\sigma)/(√(n)) \)`, and the lower control limit (LCL) is
`\( \bar{X} - A_2 * (\sigma)/(√(n)) \)`, where
`\( A_2 \)` is the control limit factor,
`\( \sigma \)` is the standard deviation, and ( n ) is the sample size. In this case, with seven observations in each of the five samples, you can calculate the control limits using the appropriate
`\( A_2 \)` value.

Step-by-step explanation:

To construct control limits for the mean
`(\(\bar{X}\))` in a process, the control limit factor
`\( A_2 \)` is crucial. For a sample size of seven
`(\( n = 7 \))`, the appropriate value for
`\( A_2 \)` is typically found in statistical tables. Once
`\( A_2 \)` is determined, you can use the formula:

`\[ UCL = \bar{X} + A_2 * (\sigma)/(√(n)) \]`

`\[ LCL = \bar{X} - A_2 * (\sigma)/(√(n)) \]`

Here,
`\( \bar{X} \)\\` is the average processing time of 10 minutes,
`\( \sigma \)` is the population standard deviation (which may be estimated from the sample standard deviation), and ( n ) is the sample size (7 in this case).

Maintaining control limits ensures that the process is within acceptable variability. If observations fall outside these limits, it suggests a potential issue with the process. This statistical approach is fundamental in quality control, allowing organizations to monitor and improve processes systematically.