Question

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

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

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.

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.

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.

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.