91Ó°ÊÓ

A luxury sports car dealership offers its clients a complimentary shuttle service to and from the dealership when their cars are serviced. Currently, the dealership's driver shuttles clients to and from locations. However, the dealership has only one driver, and clients sometimes have to wait for an extended period. In hopes of improving service and pleasing clients, the dealership decides to change from an in-house shuttle service to using a ride- sharing service. The dealership wants to monitor the impact of this change to see if the percentage of clients who take advantage of the transportation service changes. First, the dealership gathers historical data to determine the percentage of clients who have been using the shuttle service. It looks at records for the past six months. The average number of clients who visit the dealership each month is 190 , with relatively little month-to-month variation. During the past six months, a total of 354 clients have requested rides. a. What is the estimated total number of clients requesting rides during these six months? What is \(p\) ? b. Give the center line and control limits for a \(p\) chart on which to plot the future monthly proportions of clients requesting rides.

Step by step solution

01

Calculate Total Number of Clients in Six Months

To find the total number of clients who visited the dealership over the six-month period, multiply the average number of monthly clients by six: \[ \text{Total clients in 6 months} = 190 \times 6 = 1140 \] clients.

02

Estimate the Proportion of Clients Requesting Rides ( p ext{)

The proportion \( p \) of clients requesting rides is calculated by dividing the total number of clients who requested rides by the total number of clients. We have 354 clients requesting rides.\[ p = \frac{354}{1140} \approx 0.3105 \]

03

Center Line of the p ext{ Chart}

The center line for a \( p \) chart is the proportion \( p \) of clients requesting rides. Thus, the center line is \( p = 0.3105 \).

04

Calculate Standard Deviation of p ext{ ( p ext{)}

The standard deviation for proportions is given by \( \sigma = \sqrt{\frac{p(1-p)}{n}} \), where \( n \) is the average number of clients per month.\[ \sigma = \sqrt{\frac{0.3105 \times (1 - 0.3105)}{190}} \approx 0.0334 \]

05

Determine Control Limits for the p ext{ Chart}

The control limits are calculated as: \[ \text{Upper Control Limit (UCL)} = p + 3\sigma = 0.3105 + 3 \times 0.0334 \approx 0.4107 \]\[ \text{Lower Control Limit (LCL)} = p - 3\sigma = 0.3105 - 3 \times 0.0334 \approx 0.2103 \] (Ensure that the LCL is not negative).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

p-chart

P-charts are a type of control chart used in statistical process control to monitor the proportion of defective items or events in a process over time. In our example, the dealership wants to assess whether the switch to a ride-sharing service will affect the usage rate of their transportation service. A p-chart provides a visible way to track changes in these proportions, helping the dealership quickly identify any significant variations in client behavior.
By plotting the proportion of clients using the service each month on a p-chart, the dealership can assess whether there are any trends or outliers that may indicate an improvement or decline in service usage.

Proportion Calculation

Calculating the proportion is an essential first step in creating a p-chart. It involves determining the percentage of a particular outcome over the total number of opportunities. In this case, we need to find out the proportion of clients who requested rides.
Here's how we do it:

• First, calculate the total number of clients over the six-month period, which was found to be 1,140.
• Then, divide the number of clients who requested rides (354) by the total number of clients (1,140).
• The calculated proportion, \( p = \frac{354}{1140} \approx 0.3105 \), represents the percentage of clients who opted for the shuttle service.

This proportion can now be used as a foundation for creating the p-chart.

Standard Deviation

In the context of p-charts, the standard deviation measures the expected variation in the proportion of successes or features of interest (e.g., clients requesting rides). Here's how we calculate it:

• The formula for standard deviation for a proportion \( \sigma \) is given by \( \sigma = \sqrt{\frac{p(1-p)}{n}} \).
• Using our previous proportion \( p = 0.3105 \) and an average number of clients \( n = 190 \), we calculate \( \sigma = \sqrt{\frac{0.3105 \times (1 - 0.3105)}{190}} \approx 0.0334 \).

Standard deviation is crucial as it provides a sense of how much the monthly proportions might naturally fluctuate around the average. This information is vital when determining the control limits on a p-chart.

Control Limits

Control limits define the boundaries within which we expect the process proportion to fluctuate under normal conditions. They are calculated using the average proportion and its standard deviation:

• The Upper Control Limit (UCL) is calculated as \( p + 3\sigma \).
• The Lower Control Limit (LCL) is \( p - 3\sigma \), ensuring it does not fall below zero as proportions can't be negative.

In the dealership's case, with \( p = 0.3105 \) and \( \sigma = 0.0334 \):

• UCL = \( 0.3105 + 3 \times 0.0334 \approx 0.4107 \).
• LCL = \( 0.3105 - 3 \times 0.0334 \approx 0.2103 \).

These limits help determine whether the proportion of clients using the service is stable or needs further investigation if it breaches these control boundaries.