Q$1.$ Constructing a p$-$chart

The ABC Banking Corporation issues a national credit card through its various bank branches in five southeastern states. The bank credit card business is highly competitive and interest rates do not vary substantially, so the company decided to attempt to retain its customers by improving customer service through a reduction in billing errors. The credit card division monitored its billing department process by taking daily samples of $300$ customer bills for $25$ days and checking their accuracy. The sample results are as follows:

Develop a p$-$chart for the billing process using $3-$sigma $($z $=3)$ control limits and indicate if the process is out of control.

$$

Sample

Number of Defectives

$1$

$6$

$2$

$13$

$3$

$10$

$4$

$6$

$5$

$5$

$6$

$8$

$7$

$10$

$8$

$11$

$9$

$13$

$10$

$9$

$11$

$8$

$12$

$6$

$13$

$3$

$14$

$2$

$15$

$10$

$16$

$10$

$17$

$12$

$18$

$15$

$19$

$16$

$20$

$15$

$21$

$13$

$22$

$9$

$23$

$10$

$24$

$12$

$25$

$15$

$$

Q$2.$ Constructing a c$-$chart

The ABC Company daily delivers multiple orders by truck from its regional distribution center to stores in the Ted$$s Supermarket chain. One measure of its supply chain performance is the number of late deliveries. The company$$s goal is to make all deliveries within one day, so a delivery is late if it exceeds one day. The late deliveries for each of the past $20$ days are as follows:

Construct a c$-$chart for late deliveries with $3-$sigma $($z $=3)$ control limits and indicate if the delivery process was out of control at any time.

$$

Day

Late Deliveries

$1$

$6$

$2$

$15$

$3$

$16$

$4$

$8$

$5$

$19$

$6$

$12$

$7$

$10$

$8$

$15$

$9$

$9$

$10$

$9$

$11$

$6$

$12$

$12$

$13$

$16$

$14$

$10$

$15$

$15$

$16$

$18$

$17$

$14$

$18$

$12$

$19$

$19$

$20$

$22$

$$

Q$3.$ Constructing an $-$chart AND an R$-$chart

At XYZ Store, the customer service area processes customer returns, answers customer questions and provides information, addresses customer complaints, and sells gift certificates. The manager believes that if customers must wait longer than $8$ minutes to see a customer service representative, they get irritated. The customer service process has been designed to achieve a customer wait time of between $6$ and $12$ minutes. The store manager has conducted $10$ samples of six observations each of customer waiting over a two$-$week periods as follows:

Construct an $-$chart in conjunction with an R$-$chart to monitor customer service wait time, and comment on the capability of the service area to meet its designated goal.

$$

Sample

Wait Time $($min$)$

$1$

$8\mathrm{.}3$

$9\mathrm{.}6$

$10\mathrm{.}2$

$7\mathrm{.}4$

$10\mathrm{.}1$

$8\mathrm{.}5$

$2$

$8\mathrm{.}6$

$8\mathrm{.}9$

$6\mathrm{.}9$

$8\mathrm{.}3$

$9\mathrm{.}2$

$7\mathrm{.}9$

$3$

$8\mathrm{.}5$

$7\mathrm{.}8$

$12\mathrm{.}8$

$15\mathrm{.}5$

$9\mathrm{.}5$

$10\mathrm{.}9$

$4$

$6\mathrm{.}7$

$6\mathrm{.}5$

$9\mathrm{.}8$

$6\mathrm{.}5$

$8$

$8$

$5$

$11\mathrm{.}5$

$10\mathrm{.}4$

$10\mathrm{.}6$

$12\mathrm{.}7$

$11\mathrm{.}3$

$10\mathrm{.}6$

$6$

$13\mathrm{.}5$

$9\mathrm{.}2$

$9\mathrm{.}8$

$8\mathrm{.}3$

$7\mathrm{.}6$

$8\mathrm{.}6$

$7$

$15\mathrm{.}7$

$12\mathrm{.}1$

$3\mathrm{.}6$

$9\mathrm{.}5$

$13\mathrm{.}2$

$9$

$8$

$6\mathrm{.}1$

$8\mathrm{.}5$

$8\mathrm{.}7$

$6\mathrm{.}5$

$5\mathrm{.}1$

$7\mathrm{.}1$

$9$

$9\mathrm{.}3$

$12\mathrm{.}4$

$13\mathrm{.}7$

$9\mathrm{.}3$

$9\mathrm{.}2$

$11\mathrm{.}2$

$10$

$10\mathrm{.}7$

$7\mathrm{.}7$

$9$

$9\mathrm{.}1$

$11$

$12$

$$

$$

$$

Control Chart Factors for x$-$bar and R Charts

Sample size, n

Mean Factor, A$2$

Upper Range, D$4$

Lower Range, D$3$

$2$

$1\mathrm{.}88$

$3\mathrm{.}268$

$0$

$3$

$1\mathrm{.}023$

$2\mathrm{.}574$

$0$

$4$

$0\mathrm{.}729$

$2\mathrm{.}282$

$0$

$5$

$0\mathrm{.}577$

$2\mathrm{.}115$

$0$

$6$

$0\mathrm{.}483$

$2\mathrm{.}004$

$0$

$7$

$0\mathrm{.}419$

$1\mathrm{.}924$

$0\mathrm{.}076$

$8$

$0\mathrm{.}373$

$1\mathrm{.}864$

$0\mathrm{.}136$

$9$

$0\mathrm{.}337$

$1\mathrm{.}816$

$0\mathrm{.}184$

$10$

$0\mathrm{.}308$

$1\mathrm{.}777$

$0\mathrm{.}223$

$11$

$0\mathrm{.}285$

$1\mathrm{.}744$

$0\mathrm{.}256$

$12$

$0\mathrm{.}266$

$1\mathrm{.}716$

$0\mathrm{.}284$

$13$

$0\mathrm{.}249$

$1\mathrm{.}692$

$0\mathrm{.}308$

$14$

$0\mathrm{.}235$

$1\mathrm{.}671$

$0\mathrm{.}329$

$15$

$0\mathrm{.}223$

$1\mathrm{.}652$

$0\mathrm{.}348$

$16$

$0\mathrm{.}212$

$1\mathrm{.}636$

$0\mathrm{.}364$

$17$

$0\mathrm{.}203$

$1\mathrm{.}621$

$0\mathrm{.}379$

$18$

$0\mathrm{.}194$

$1\mathrm{.}608$

$0\mathrm{.}392$

$19$

$0\mathrm{.}187$

$1\mathrm{.}596$

$0\mathrm{.}404$

$20$

$0\mathrm{.}180$

$1\mathrm{.}586$

$0\mathrm{.}414$