To sketch a modified boxplot for the data provided, follow these steps:

Data:

2, 3, 4, 5, 5, 8, 8, 13, 13, 13, 15, 17, 19, 21, 21, 21, 25, 25, 47, 55

Steps for Modified Boxplot:

1. Order the Data: The data is already ordered.

2. Identify the Five-Number Summary:

   • Minimum: $2$
   • First Quartile $Q_1$ (25th percentile): The median of the lower half of the data = $8$
   • Median $Q_2$ (50th percentile): Middle of the dataset = $15$
   • Third Quartile $Q_3$ (75th percentile): The median of the upper half of the data = $25$
   • Maximum: $55$

3. Identify Outliers:

   • Interquartile Range (IQR): $Q_3 - Q_1 = 25 - 8 = 17$
   • Outlier Thresholds:
     • Lower threshold: $Q_1 - 1.5 \times IQR = 8 - 1.5(17) = -17.5$ (No lower outliers since all values are above 0)
     • Upper threshold: $Q_3 + 1.5 \times IQR = 25 + 1.5(17) = 50.5$
   • Any data points greater than $50.5$ will be considered outliers. In this case, $55$ is an outlier.

4. Boxplot Construction:

   • Draw a number line that covers the range of your data (e.g., 0 to 60).
   • Draw a box from $Q_1 = 8$ to $Q_3 = 25$.
   • Draw a line at the median $Q_2 = 15$.
   • Draw "whiskers" from $Q_1 = 8$ to the smallest non-outlier $2$, and from $Q_3 = 25$ to the largest non-outlier $47$.
   • Plot $55$ as a single point beyond the upper whisker, marking it as an outlier.

5. Label the Axes:

   • X-axis: Label as "Length of Stay (in days)".
   • Title: "Modified Boxplot of Length of Stay for 20 Patients".

Next Step:

Please sketch this boxplot based on the steps outlined and upload the image for review.

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Would you like any further details or clarifications? Here are 5 relative questions to expand on this:

1. How do you interpret outliers in a modified boxplot?
2. What are some common applications of boxplots in healthcare statistics?
3. Can you calculate the mean length of stay from the dataset provided?
4. How does a modified boxplot differ from a standard boxplot?
5. What is the impact of skewed data on the shape of a boxplot?

Tip: Always double-check if your dataset contains outliers, as they can significantly affect interpretations of your results!