91Ó°ÊÓ

The manager of Motel 11 has 316 rooms in Palo Alto, California. From observation over a long period of time, she knows that on an average night, 268 rooms will be rented. The long-term standard deviation is 12 rooms. This distribution is approximately mound-shaped and symmetric. (a) For 10 consecutive nights, the following numbers of rooms were rented each night: $$\begin{array}{l|cccccc} \hline \text { Night } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Number of rooms } & 234 & 258 & 265 & 271 & 283 & 267 \\ \hline & & & & & & \\ \hline \text { Night } & 7 & 8 & 9 & 10 & & \\ \hline \text { Number of rooms } & 290 & 286 & 263 & 240 & & \\ \hline \end{array}$$ Make a control chart for the number of rooms rented each night, and plot the preceding data on the control chart. Interpretation Looking at the control chart, would you say the number of rooms rented during this 10-night period has been unusually low? unusually high? about what you expected? Explain your answer. Identify all out-of-control signals by type (I, II, or III). Make a control chart for the number of rooms rented each night, and plot the preceding data on the control chart. Interpretation Looking at the control chart, would you say the number of rooms rented during this 10-night period has been unusually low? unusually high? about what you expected? Explain your answer. Identify all out-of-control signals by type (I, II, or III). (b) For another 10 consecutive nights, the following numbers of rooms were rented each night: $$\begin{array}{l|cccccc} \hline \text { Night } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Number of rooms } & 238 & 245 & 261 & 269 & 273 & 250 \\ \hline & & & & & & \\ \hline \text { Night } & 7 & 8 & 9 & 10 & & \\ \hline \text { Number of rooms } & 241 & 230 & 215 & 217 & & \\ \hline \end{array}$$ Make a control chart for the number of rooms rented each night, and plot the preceding data on the control chart. Would you say the room occupancy has been high? low? about what you expected? Explain your answer. Identify all out-of- control signals by type (I, II, or IIII).

Step by step solution

01

Determine Control Limits

Calculate the control limits for the control chart. The center line will be the average number of rooms rented, which is 268. The upper control limit (UCL) is given by \(UCL = \mu + 3\sigma\), where \(\mu = 268\) and \(\sigma = 12\). So, \(UCL = 268 + 36 = 304\). The lower control limit (LCL) is \(LCL = \mu - 3\sigma = 268 - 36 = 232\).

02

Plot Night 1-10 Data on Control Chart

Plot the following room numbers on a control chart with control limits obtained in Step 1: 234, 258, 265, 271, 283, 267, 290, 286, 263, 240. All points should fall within the control limits 232 and 304, indicating stability in the process.

03

Evaluate Night 1-10 Data for Out-of-Control Signals

Look for any patterns or points outside the control limits. For Nights 1-10, all are within limits, but Night 1 (234) and Night 10 (240) are close to the LCL, indicating a potential signal Type II (run near a control limit).

04

Interpretation for Night 1-10

The occupancy for these 10 nights appears about as expected as all points are within the control limits and there are no major patterns indicating a shift. The system is stable.

05

Plot Night 11-20 Data on Control Chart

For the next set of data (238, 245, 261, 269, 273, 250, 241, 230, 215, 217), plot these numbers on the same control chart. Note that the number 215 and 217 on Nights 19 and 20 fall below the LCL of 232.

06

Evaluate Night 11-20 Data for Out-of-Control Signals

The points for Nights 19 and 20 are out-of-control since they fall below the LCL. This indicates a signal Type I (point outside control limits). This suggests periods of unusually low occupancy.

07

Interpretation for Night 11-20

The room occupancy from Nights 11-20 is lower than expected, with two nights registering below the control limit, indicating an out-of-control process with unusually low room rental numbers.

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Key Concepts

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

Control Limits

Control limits are essential components of a control chart and play a significant role in the quality control process. They help in determining whether a process is stable or has gone out of control. The control limits are calculated using the average (\( \mu \)) and the standard deviation (\( \sigma \)) of the process data.

The center line in a control chart represents the average expected value, serving as a reference point for the data. For instance, in the case of the Motel 11, with an average room rental of 268, this value becomes the center line. The upper control limit (UCL) and lower control limit (LCL) are set at three standard deviations higher and lower than this average, respectively. This is calculated as:

• UCL = \( \mu + 3\sigma = 268 + 36 = 304 \)
• LCL = \( \mu - 3\sigma = 268 - 36 = 232 \)

The control limits serve as benchmarks to measure the data against, helping to identify any points that fall significantly outside the expected range, signaling potential issues with process stability.

Out-of-Control Signals

Out-of-control signals are critical alerts indicating that a process may not be behaving as expected. These signals are detected when data points fall outside the control limits or when specific patterns emerge within the control chart.

There are different types of out-of-control signals, such as:

• Type I: A single point outside the control limits, indicating an extreme deviation from the normal range.
• Type II: Several points consistently near a control limit, suggesting a potential systemic issue approaching a threshold.
• Type III: Recognizable patterns within the control limits, like a run or trend, which may indicate a shift in the process mean.

In the analysis of Motel 11's occupancy rates, during Nights 1-10, some signals indicated potential issues, such as Nights 1 and 10 being close to the LCL. For Nights 11-20, Nights 19 and 20 displayed Type I out-of-control signals by falling below the LCL, indicating unusual low occupancy.

Statistical Process Control

Statistical Process Control (SPC) involves using statistical methods to monitor and control a process. This analysis helps in maintaining process quality by identifying variations and implementing corrective measures when necessary.

SPC employs control charts to visually track process behavior over time. For example, monitoring room occupancy rates at Motel 11 involves plotting daily room rentals on the control chart and comparing them against control limits derived from historical data. This approach enables the detection of any deviations indicating problems or necessary improvements.

By regularly applying SPC, businesses can ensure that their operations are in control, reducing variability and enhancing efficiency. It assists in predicting future performance and helps in making data-driven decisions to optimize processes.

Room Occupancy Rates

Room occupancy rates are a critical performance metric in the hospitality industry, indicating how effectively a hotel utilizes its available rooms. These rates are calculated by dividing the number of rooms rented by the total available rooms and are represented as a percentage.

For Motel 11, examining room occupancy rates for consecutive nights involved plotting rental data on a control chart to assess stability. Observing trends in this data can highlight periods of low or high demand and help in strategic decision-making.

Maintaining consistent room occupancy rates close to the expected average can indicate stability and effective management. For instance, during Nights 1-10, all rental data fell within control limits, demonstrating a normal level of occupancy. However, Nights 11-20 showed a decline with several nights below the lower control limit, suggesting an opportunity to investigate and improve strategies to increase occupancy rates.