To make the median of the new data equal to 20, additional observations of a value greater than or equal to 20 must be added. Let's analyze step by step:

1. Initial Setup

The given observations are: $14, 5, 13, 14, 8, 8, 0, 18, 10$

• Total observations: $n = 9$
• Current sorted data: $0, 5, 8, 8, 10, 13, 14, 14, 18$
• Current median: The 5th observation (middle value for $n = 9$) is 10.

2. Target Median

For the median to be $20$, at least half of the sorted dataset must have values $\geq 20$. This means the position of $20$ in the sorted list should correspond to the middle position when we add new observations.

If $k$ new observations are added, the total number of observations becomes $n + k$. The median position will then be: $\text{Median position} = \frac{n + k + 1}{2}$

3. Condition for Median ≥ 20

To ensure the median is $20$, at least $\frac{n + k + 1}{2}$ observations must have values $\geq 20$.

4. Adding Observations

• Current observations $< 20$: $0, 5, 8, 8, 10, 13, 14, 14, 18$ — all 9 values.
• Observations $\geq 20$: $0$ (none exist in the current data).

We need at least $\frac{9 + k + 1}{2}$ observations $\geq 20$. Since all added observations will be $20$ or greater, the inequality becomes: $\frac{9 + k + 1}{2} \leq k$

Simplify: $\frac{10 + k}{2} \leq k$

Multiply through by 2: $10 + k \leq 2k$

Solve for $k$: $k \geq 10$

5. Conclusion

At least 10 new observations of $20$ or greater must be added to make the median $20$.

Would you like a detailed explanation of these steps or related questions? Here are some questions for further exploration:

1. How does adding fewer observations affect the median?
2. How can the mean and mode of the data be influenced by adding observations?
3. What is the effect on the dataset's range by adding new observations?
4. How is the position of the median calculated for even-sized datasets?
5. Could the median be made any value other than $20$ by adding different observations?

Tip: For quick calculations of medians in sorted data, always focus on the middle position formula: $\frac{n+1}{2}$.