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Chapter 3: Problem 2

Data on tipping percent for 20 restaurant tables, consistent with summary statistics given in the paper "Beauty and the Labor Market: Evidence from Restaurant Servers" (unpublished manuscript by Matt Parrett, 2007 ), are: $$ \begin{array}{rrrrrrrr} 0.0 & 5.0 & 45.0 & 32.8 & 13.9 & 10.4 & 55.2 & 50.0 \\ 10.0 & 14.6 & 38.4 & 23.0 & 27.9 & 27.9 & 105.0 & 19.0 \\ 10.0 & 32.1 & 11.1 & 15.0 & & & & \end{array} $$

Short Answer

Expert verified
The mean tipping percent for the 20 restaurant tables is 25.965%, and the standard deviation is approximately 19.55%. This indicates a significant variability in the tips across these tables.

Step by step solution

01

Calculating the mean

To calculate the mean tipping %, start by adding up all the tipping percentages together and then dividing by the total number of tables, which in this case is 20. \( Mean = \frac{\sum{tipping \ percentages}}{Number \ of \ tables} \)
02

Calculating the mean value

Add all the tipping percentages together and divide by 20: \( Mean = \frac{0.0 + 5.0 + 45.0 + 32.8 + 13.9 + 10.4 + 55.2 + 50.0 + 10.0 + 14.6 + 38.4 + 23.0 + 27.9 + 27.9 + 105.0 + 19.0 + 10.0 + 32.1 + 11.1 + 15.0}{20} \) \( Mean = \frac{519.3}{20} \) \( Mean = 25.965 \) So, the mean tipping % for these 20 tables is 25.965%.
03

Calculating the Standard Deviation

To calculate the standard deviation, we first calculate the variance which is the average of the squared differences from the mean. Follow the steps below: 1. Find the difference between each tipping percentage and the mean. 2. Square these differences. 3. Add all the squared differences together. 4. Divide the sum of squared differences by the number of tables (20) to find the variance. 5. Then, find the square root of the variance to calculate the standard deviation.
04

Calculating the Variance

1. Calculate the squared differences between each tipping percentage and the mean (25.965). \( (0.0 - 25.965)^2 = 674.632225 \) (similarly find squared differences for all tipping %) 2. Add all the squared differences together: \( Sum \ of \ squared \ differences = 674.632225 + ... + 121.262225 = 7644.06525 \) 3. Divide the sum of squared differences by the number of tables (20): \( Variance = \frac{7644.06525}{20} = 382.2032625 \)
05

Calculating the Standard Deviation

Now, find the square root of the variance: \( Standard \ Deviation = \sqrt{382.2032625} = 19.54803 \) So, the standard deviation of tipping % for these 20 tables is approximately 19.55%. In conclusion, the data on tipping percent for these 20 restaurant tables has a mean tipping % of 25.965% and a standard deviation of 19.55%. This indicates that there is quite a bit of variability in the tips across these tables.

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Key Concepts

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

Mean Calculation
Understanding the concept of "mean" is essential in descriptive statistics. The mean, or average, is a common measure of central tendency, which represents the central point of a data set. To find the mean tipping percentage from the given restaurant data, we first add up all the tipping values. This operation provides the total amount of all tips combined across 20 tables. To find the actual mean, we divide this total by the number of restaurant tables.

Here's a simplified breakdown of the calculation process:
  • Add all given tipping percentages together.
  • Divide the calculated sum by 20, the number of tables in the sample.
This operation results in a mean tipping percentage of 25.965%. This value helps provide an average tipping trend in the restaurant setting, summarizing the behavior of all tables with just one number.
Standard Deviation
Standard deviation is a statistical measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the data points tend to be close to the mean, whereas a high standard deviation signifies that the data are spread out over a wider range.

To find the standard deviation of the tipping percentages, we go through these key steps:
  • Calculate the variance first by taking each tipping percentage, subtract the mean from it, and square the result.
  • Add all these squared deviations together to get the sum of squared differences.
  • Divide this sum by the total number of data points, which is 20, to find the variance.
Finally, we calculate the square root of this variance to find the standard deviation, which is approximately 19.55 in this data set. A relatively high standard deviation, like in this example, suggests significant variability in tipping percentages among restaurant tables.
Variance
Variance plays a crucial role in understanding data variability, as it measures how much the numbers in a data set differ from the average (mean) of those numbers. It is essentially the average of the squared differences from the mean.

We calculate the variance by following these steps:
  • Compute the difference between each tipping percentage and the mean (25.965%).
  • Square these differences to get rid of negative values and emphasize larger deviations.
  • Sum all those squared differences.
  • Finally, divide this total by the number of observations (20 tables) in the data set.
For this specific case, the variance is calculated to be 382.2032625. Variance helps statisticians and researchers understand how data points (in this case, tipping percentages) are distributed in relation to the mean. When combined with standard deviation, it paints a more comprehensive picture of data distribution.

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