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Fast-food restaurants spend quite a bit of time studying the amount of time cars spend in their drive-through. Certainly, the faster the cars get through the drive-through, the more the opportunity for making money. QSR Magazine studied drive-through times for fast-food restaurants, and found Wendy's had the best time, with a mean time a car spent in the drive-through equal to 138.5 seconds. Assume that drive-through times are normally distributed, with a standard deviation of 29 seconds. Suppose that Wendy's wants to institute a policy at its restaurants that it will not charge any patron that must wait more than a certain amount of time for an order. Management does not want to give away free meals to more than \(1 \%\) of the patrons. What time would you recommend Wendy's advertise as the maximum wait time before a free meal is awarded?

Step by step solution

01

Understand the Problem

We need to find the maximum time (in seconds) a car can spend in the drive-through before Wendy's gives a free meal to no more than 1% of patrons.

02

Define the Distribution

The drive-through times are normally distributed with a mean (\(\text{mean} = 138.5\) seconds) and a standard deviation (\(\text{sd} = 29\) seconds).

03

Identify the Percentile

We need to find the time corresponding to the 99th percentile of the normal distribution, since 1% of patrons should get free meals due to waiting time.

04

Use the Z-Score Formula

The formula to convert a percentile to a time is:\[X = \text{mean} + (Z \times \text{sd})\]Where \(Z\) corresponds to the 99th percentile. From Z-tables or a standard normal distribution calculator, \(Z_{99\text{th percentile}}\) = 2.33.

05

Calculate the Maximum Time

Using the Z-score of 2.33, we have:\[X = 138.5 + (2.33 \times 29) = 138.5 + 67.57 = 206.07 \text{ seconds}\]

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Key Concepts

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

Normal Distribution

In business and statistics, many processes and measurements follow a pattern called the normal distribution. It is often represented as a bell-shaped curve. Most values cluster around a central mean, and the probability of values decreases as you move away from the mean. For Wendy's drive-through times, the mean is 138.5 seconds with a standard deviation of 29 seconds. This means most customers experience a waiting time close to 138.5 seconds, and fewer will have times much lower or higher.

Percentile Calculation

Percentiles help you understand the relative standing of a value within a dataset. If we say a value is at the 99th percentile, it means 99% of the data points fall below this value. For Wendy's, we want to set a waiting time limit such that only the top 1% of waiting times (that might qualify for a free meal) exceed this limit. This means finding the 99th percentile of the normal distribution of waiting times.

Z-Score

Z-scores are a standardized way of describing where a value lies within a normal distribution. The formula for calculating the z-score is:o\[ Z = \frac{(X - \text{mean})}{\text{sd}} \]o\where **X** is the value you are standardizing, **mean** is the average of the dataset, and **sd** is the standard deviation. To reverse this and find a specific value from a z-score and percentile, you use:o\[ X = \text{mean} + (Z \times \text{sd}) \]o FFor the 99th percentile, **Z** = 2.33, and thus for Wendy's:o\[ X = 138.5 + (2.33 \times 29) \approx 206.07 \text{ seconds} \]

Business Applications of Statistics

Statistics play a crucial role in aiding decision-making in business. Fast-food chains like Wendy's use it to streamline operations and improve customer satisfaction. By analyzing waiting times with normal distribution and determining specific percentiles, businesses can make data-informed decisions. This helps set achievable goals and service standards—like not charging patrons whose waiting time exceeds a certain threshold—which can enhance customer experience and operational efficiency.

Waiting Time Analysis

Analyzing waiting times is vital for service-based businesses. Long waiting times can lead to customer dissatisfaction and lost revenue. By understanding the distribution of waiting times, businesses can pinpoint areas needing improvement. For Wendy's, setting a wait time policy based on the top 1% of distribution helps minimize the number of free meals while keeping wait times practical and acceptable for most customers. This analysis ensures they can balance service quality with operational costs effectively.