how to statistically compare random investment returns to actual returns

I have a string of (344) investment returns from an investment strategy. I compared these daily returns (my actual picks) to returns from random investment picks from the same stock universe. My returns won 60% of the days. A friend claims this is not that good as you would expect the random returns to win 50% of the time, so I really only won 10% of the time. How do I dispute this?

Here' my thinking: I understand that normal distributions (the distribution I presume for investment returns allowing both long and short positions) are 50/50 around the mean. To presume a random sample would beat mine 50% of the time implies consistent distributions, i.e., same mean/variance structure.

I think one answer is to perform a paired t-test with null hypothesis of no difference in means. A high enough t would prove random should not beat mine 50/50. 1. Does that make sense? 2. Is there something better?

Any help/thoughts would be appreciated. Thanks