A financial analyst is testing a new algorithm that predicts stock price movements. He hypothesizes that the mean change in stock prices after implementing the algorithm differs from zero. To test this hypothesis, he collects a sample of 50 trading days and calculates the sample mean change, which is $3.5$, and the sample standard deviation, which is $6.0$. The analyst will conduct a two-tailed test at a significance level of $0.05$.
Using the appropriate test statistic, what is the analyst's calculated z-value for the hypothesis test?