The given problem involves understanding the nature of a probability distribution. Here is a detailed, step-by-step analysis to determine whether the distribution is symmetric, skewed, or both:
1. Understanding the Data:
We are given probabilities for different values of [tex]\( k \)[/tex], where [tex]\( k \)[/tex] is the number of adults (out of 6) who watch a particular television show.
[tex]\[
\begin{array}{r}
P(k=0)=0.0467 \\
P(k=1)=0.1866 \\
P(k=2)=0.3110 \\
P(k=3)=0.2765 \\
P(k=4)=0.1382 \\
P(k=5)=0.0369
\end{array}
\][/tex]
2. Plotting the Probabilities:
To understand the shape of the distribution, imagine plotting these probabilities on a graph with [tex]\( k \)[/tex] on the x-axis and [tex]\( P(k) \)[/tex] on the y-axis.
[tex]\[
\begin{array}{c|c}
k & P(k) \\
\hline
0 & 0.0467 \\
1 & 0.1866 \\
2 & 0.3110 \\
3 & 0.2765 \\
4 & 0.1382 \\
5 & 0.0369 \\
\end{array}
\][/tex]
3. Interpreting the Probabilities:
- For [tex]\( k=0 \)[/tex], [tex]\( P=0.0467 \)[/tex]
- For [tex]\( k=1 \)[/tex], [tex]\( P=0.1866 \)[/tex]
- For [tex]\( k=2 \)[/tex], [tex]\( P=0.3110 \)[/tex]
- For [tex]\( k=3 \)[/tex], [tex]\( P=0.2765 \)[/tex]
- For [tex]\( k=4 \)[/tex], [tex]\( P=0.1382 \)[/tex]
- For [tex]\( k=5 \)[/tex], [tex]\( P=0.0369 \)[/tex]
4. Analyzing the Shape:
- Notice that [tex]\( P(k) \)[/tex] increases from [tex]\( k=0 \)[/tex] to [tex]\( k=2 \)[/tex].
- From [tex]\( k=2 \)[/tex] to [tex]\( k=3 \)[/tex], it starts decreasing.
- As [tex]\( k \)[/tex] further increases from 3 to 5, [tex]\( P(k) \)[/tex] continues to decrease.
The probabilities rise to a peak at [tex]\( k=2 \)[/tex] and then gradually decline.
5. Determining Symmetry or Skewness:
- Symmetric distributions have their peak (or median) in the center and decline symmetrically on both sides.
- Here, the peak is at [tex]\( k=2 \)[/tex], but the decline from [tex]\( k=2 \)[/tex] to [tex]\( k=5 \)[/tex] suggests a skewed distribution.
- Specifically, the distribution appears to decrease more gradually on the right-side, suggesting it is skewed. The longer tail on the right indicates left-skew (negative skew).
Thus, we conclude that the distribution is skewed, and specifically, this skewness implies that it is the left-skewed distribution.
Therefore, the result is:
The distribution is skewed.
