To find the median of a probability distribution, we need to determine the value at which the cumulative distribution reaches or exceeds 0.5. Here is a step-by-step solution:
1. Identify the given probabilities:
- The table provides the probabilities for the number of free throws made, ranging from 0 to 8.
[tex]$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline \text{Number of Free} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\text{Throws Made} & & & & & & & & & \\
\hline \text{Probability} & 0.002 & 0.008 & 0.04 & 0.12 & 0.23 & 0.28 & 0.21 & 0.09 & 0.02 \\
\hline
\end{array}$[/tex]
2. Calculate the cumulative probabilities:
- Cumulative probability is the sum of the probabilities up to and including each value.
[tex]$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline \text{Number of Free} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\text{Throws Made} & & & & & & & & & \\
\hline \text{Cumulative Probability} & 0.002 & 0.01 & 0.05 & 0.17 & 0.4 & 0.68 & 0.89 & 0.98 & 1.0 \\
\hline
\end{array}$[/tex]
3. Locate the median:
- The median is found where the cumulative probability first reaches or exceeds 0.5.
- From the cumulative probabilities above:
- For 0: 0.002
- For 1: 0.01
- For 2: 0.05
- For 3: 0.17
- For 4: 0.4
- For 5: 0.68 (this is the first value that exceeds 0.5)
- For 6: 0.89
- For 7: 0.98
- For 8: 1.0
The cumulative probability exceeds 0.5 at a value of 5 free throws.
Therefore, the median of the distribution for the number of free throws made is [tex]\( \boxed{5} \)[/tex].
