How Does Removing Zeros Affect Sharpe?

Consider a return series with $2n$ observations. Exactly $n$ of them are zero, and the other $n$ are non-zero returns $x_1, x_2, \dots, x_n$ with mean $\mu$ and standard deviation $\sigma$. Take the risk-free rate to be $0$, so the Sharpe ratio is given by $\text{Sharpe} = \frac{\text{mean(return)}}{\text{std(return)}}$

1. Compare the Sharpe ratio based on all $2n$ observations with the Sharpe ratio based only on the $n$ non-zero observations.
2. Is the resulting Sharpe ratio higher or lower?
3. If the answer depends on the sign of the mean return, explain why.
4. Briefly discuss whether the conclusion still holds if the $n$ removed observations are not necessarily the zero-return ones.

Overview: This question tests understanding of the Sharpe ratio and how changing the sample composition by dropping zero-return observations affects the sample mean and standard deviation, and therefore risk-adjusted performance measures.