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Daily Tip Revenue for a Waitress Data 2.12 on page 123 describes information from a sample of 157 restaurant bills collected at the First Crush bistro. The data is available in RestaurantTips. Two intervals are given below for the average tip left at a restaurant; one is a \(90 \%\) confidence interval and one is a \(99 \%\) confidence interval. Interval A: 3.55 to 4.15 Interval B: 3.35 to 4.35 (a) Which one is the \(90 \%\) confidence interval? Which one is the \(99 \%\) confidence interval? (b) One waitress generally waits on 20 tables in an average shift. Give a range for her expected daily tip revenue, using both \(90 \%\) and \(99 \%\) confidence. Interpret your results.

Short Answer

Expert verified
The 90% confidence interval is 3.55 to 4.15 (Interval A) and the 99% confidence interval is 3.35 to 4.35 (Interval B). If the waitress waits on 20 tables, we are 90% confident that her daily tip revenue would be between $71 and $83 and 99% confident that it would be between $67 and $87.

Step by step solution

01

Understanding Confidence Intervals

The confidence interval with a smaller range is always the one with the lower confidence level. Hence, Interval A (3.55 to 4.15) corresponds to the 90% confidence interval, and Interval B (3.35 to 4.35) corresponds to the 99% confidence interval. This is because a 90% confidence interval will be more narrow compared to a 99% confidence interval. The 99% confidence interval has to account for more variability, thus, it is wider in range.
02

Calculating Expected Daily Tip Revenue - 90% Confidence Interval

First, consider the 90% confidence interval (3.55 to 4.15). This means that we can be 90% confident that the average tip left at a restaurant falls within this range. If a waitress serves 20 tables in a day, her expected daily tip revenue will fall between \(3.55 \times 20 = 71\) and \(4.15 \times 20 = 83\), so her tip revenue will be in the range of 71 to 83 dollars.
03

Calculating Expected Daily Tip Revenue - 99% Confidence Interval

Next, consider the 99% confidence interval (3.35 to 4.35). This means that we can be 99% confident that the average tip left at a restaurant falls within this range. If the same waitress serves 20 tables in a day, her expected daily tip revenue will be between \(3.35 \times 20 = 67\) and \(4.35 \times 20 = 87\), so this time her tip revenue will lie in the range of 67 to 87 dollars.
04

Interpretation of Results

What these calculations represent is that if the waitress waits on 20 tables in an average shift, then we are 90% confident that she will earn between $71 and $83 in tips, and 99% confident that she will earn between $67 and $87 in tips. The larger range in the 99% confidence interval accounts for more variability in tip amounts.

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Key Concepts

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

Average Tip Revenue
Understanding the average tip revenue is essential to evaluate how much a waitress can expect to earn in tips on any given day. When analyzing tip revenue, we look at a range of values that represent the tips customers leave. The average is then determined by dividing the total collected tips by the number of transactions (bills).

For instance, if from 157 restaurant bills, the tips collected summed up to a total amount, we calculate the average tip by dividing this sum by 157. This gives us a baseline figure that indicates the average tip left per bill at the restaurant.
  • This average helps in understanding the general tipping behavior at that particular restaurant.
  • It sets the groundwork for further calculations, specifically by using confidence intervals to predict expected revenues.
Mastery of average tip revenue allows restaurant managers and staff to set realistic earning expectations and can guide decisions on staffing and operational hours.
Expected Value Calculation
Expected value calculations are a powerful tool in decision-making, particularly when trying to predict potential earnings. In the context of a waitress serving at a restaurant, expected value helps her estimate how much she might earn in tips over a given period.

Let's dive into how you can calculate expected daily tip revenue:
  • First, identify the average tip range, for instance, from a confidence interval like the 90% confidence interval with bounds $3.55 to $4.15 per table.
  • If the waitress serves 20 tables, multiply the range bounds by 20 to get the total expected tip revenue: lower bound $71 and upper bound $83.
  • The same approach applies to a 99% confidence interval with bounds $3.35 to $4.35, leading to an expected range of $67 to $87 in tips.
These calculations are essential because they provide realistic figures that the waitress can rely on for budgeting and financial planning. Additionally, they give insights into the variance in potential earnings depending on the confidence level used in the calculations.
Restaurant Data Analysis
Restaurant data analysis involves examining various metrics to improve business operations and revenue strategies. When analyzing tips, it provides valuable insights into customer behavior and staff performance.

Here’s why analysis of tip data is significant:
  • It helps detect patterns in tipping, such as peak times when people are more generous, which can guide staff scheduling.
  • Differences in tipping by different waitstaff may point to performance variations or indicate customer satisfaction issues.
  • Such analysis can also foresee trends in customer visits and help in designing promotional campaigns or special offers.
By leveraging this data effectively, restaurant managers can enhance service quality, optimize staff routines, and increase overall customer satisfaction and profits. Data analysis serves as a fundamental asset in transforming raw numbers into actionable business strategies.

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