2018-I-11

1. Since $k$ is a positive integer, then $k > 0$. The least possible value of $k$ is $1$.
2. Since the mode of the distribution is $2$, then the frequencies of other data must less than $9$. The greatest possible value of $k$ is $8$.

1. If $k \le 2$, the data at the middle will become $3$. Hence, the least possible value of $k$ is $3$.
2. If $k \ge 20$, the data at the middle will become $1$. Hence, the least possible value of $k$ is $19$.

$\begin{array}{rcl}
\dfrac{0\times k + 1 \times 2 + 2 \times 9 + 3 \times 6 + 4 \times 7}{K + 2 + 9 + 6 + 7} & = & 2\\
\dfrac{66}{k+24} & = & 2 \\
2k + 48 & = & 66 \\
k & = & 9
\end{array}$