Part 1: Quartile Calculation and Interpretation

We will first calculate the quartiles for the given dataset, and then interpret them according to the problem's structure.

The data set is: $6, 15, 3, 6, 5, 29, 7, 5, 25, 9, 8, 45, 11, 11, 33, 0, 28, 6, 9, 5, 19, 8, 22, 9, 30, 16, 55, 2, 11, 29, 0, 2, 5, 10, 3, 8, 5, 13, 7, 16$

To find the quartiles, follow these steps:

1. Arrange the data in increasing order: $0, 0, 2, 2, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 11, 11, 11, 11, 13, 15, 16, 16, 19, 22, 25, 28, 29, 29, 30, 33, 45, 55$

2. Number of data points: There are 42 values.

3. Quartiles:

   • First Quartile (Q1): The 25th percentile. This is at position $\frac{42+1}{4} = 10.75$. So, take the average of the 10th and 11th values: $Q1 = \frac{5 + 5}{2} = 5$
   • Second Quartile (Q2 or Median): The 50th percentile. This is at position $\frac{42+1}{2} = 21.5$. So, take the average of the 21st and 22nd values: $Q2 = \frac{9 + 9}{2} = 9$
   • Third Quartile (Q3): The 75th percentile. This is at position $\frac{3(42+1)}{4} = 32.25$. So, take the average of the 32nd and 33rd values: $Q3 = \frac{16 + 19}{2} = 17.5$

4. Interpretation:

   • About 25% of the wait times are 5 minutes or less.
   • About 75% of the wait times exceed 5 minutes.
   • About 50% of the wait times are 9 minutes or less.
   • About 50% of the wait times exceed 9 minutes.
   • About 75% of the wait times are 17.5 minutes or less.
   • About 25% of the wait times exceed 17.5 minutes.

Part 2: Outlier Detection

To detect outliers, we will use the Interquartile Range (IQR): $IQR = Q3 - Q1 = 17.5 - 5 = 12.5$ Now, calculate the boundaries for outliers:

• Lower bound: $Q1 - 1.5 \times IQR = 5 - 1.5 \times 12.5 = 5 - 18.75 = -13.75$ (no negative outliers since all data is non-negative).
• Upper bound: $Q3 + 1.5 \times IQR = 17.5 + 1.5 \times 12.5 = 17.5 + 18.75 = 36.25$.

Any data points greater than 36.25 are considered outliers. From the data set, the outliers are: $45, 55$

Conclusion for Part 2:

The outliers in the data set are 45 and 55.

Would you like more details or further clarifications? Here are some related questions to deepen your understanding:

1. How would you calculate the quartiles for a different dataset?
2. Why is the Interquartile Range (IQR) used for detecting outliers?
3. Can a dataset have more than two outliers? How would you handle such a case?
4. How do quartiles help in understanding the spread of data?
5. How does changing the dataset affect the calculation of quartiles and outliers?

Tip: Always double-check data order when calculating quartiles; an incorrect order can lead to errors in your calculations.