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 $19$.

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