You are a Quality Manager for an Operations Facility that makes widgets for the parent company World Class Widgets. The Quality Assurance process measures the OD of $1$ widget per day. Please use the following data set to answer the homework questions below.

Date Measurement

$9/1/20223\mathrm{.}10$

$9/2/20223\mathrm{.}00$

$9/3/20223\mathrm{.}15$

$9/4/20223\mathrm{.}20$

$9/5/20223\mathrm{.}25$

$9/6/20223\mathrm{.}18$

$9/7/20222\mathrm{.}95$

$9/8/20222\mathrm{.}50$

$9/9/20223\mathrm{.}00$

$9/10/20223\mathrm{.}08$

$9/11/20223\mathrm{.}02$

$9/12/20223\mathrm{.}05$

$9/13/20223\mathrm{.}10$

$9/14/20223\mathrm{.}11$

$9/15/20223\mathrm{.}07$

$9/16/20223\mathrm{.}21$

$9/17/20223\mathrm{.}20$

$9/18/20223\mathrm{.}15$

$9/19/20223\mathrm{.}12$

$9/20/20223\mathrm{.}18$

$9/21/20223\mathrm{.}08$

$9/22/20223\mathrm{.}02$

$9/23/20223\mathrm{.}08$

$9/24/20223\mathrm{.}22$

$1.)$ What is the mean of this data set?

$2.)$ What is the standard deviation of this data set?

$3.)$ Create a Control Chart in Excel from the data set. How many days are Out of Control $($outside of the Control Limits$)?$

$4.)$ Create a Control Chart in Excel from the data set. What is the Upper Control Limit $($UCL$)?$

$5.)$ Create a Control Chart in Excel from the data set. What is the Lower Control Limit $($LCL$)?$

$6.)$ Create a Process Capability Analysis. The Upper Spec Limit $($USL$)$ is $3\mathrm{.}30$ and the Lower Spec Limit $($LSL$)$ is $2\mathrm{.}80.$

What is the Process Capability Potential $($Cp$)?$

$7.)$ Create a Process Capability Analysis. The Upper Spec Limit $($USL$)$ is $3\mathrm{.}30$ and the Lower Spec Limit $($LSL$)$ is $2\mathrm{.}80.$

What is the Current State Process Capability $($Cpk$)?$

$8.)$ You go out to investigate any out of control dates to ensure nothing is wrong with the process and try to find a root cause for this. After talking to the operators you realize that the data point of $"2\mathrm{.}5"$ was manually entered incorrectly and the actual reading was $"3\mathrm{.}2".$

Re$-$evaluate the process capability with this error corrected. What is the updated$/$new Current State Process Capability $($Cpk$)?($Add screenshots of the excel graphs$)$