I am trying to compare the hours worked during a 3 month time frame for two groups of people, male and female. What I want to tell is if there is significant difference between the average total hours worked for men and the average total hours worked for women.

Is a t-test appropriate here? If I get a t Stat of -1.36 and a t Critical of 1.68 this should tell me there is not a significant difference. Or am I way off here?

Would I also be able to use a z-score in this situation? If my math is correct I can calculate the z-score by taking the mean from the female group minus the mean from the male group and divide that by the standard deviation of the female group to determine how many standard deviations away from the male mean the female mean is. So (58.9-77.7)/37.2= -0.505 The z-score here would tell me the female mean is half a standard deviation below the male mean which would not be significant. Is my thinking correct on this?

Thanks in advance for any help or leads in the right direction.

You are correct. The t-test is most commonly used outside of an artificial, academic situation. The z-test should only be used when the population standard deviation is known and the sample size > 30. The population standard deviation is rarely known, which limits the use of this test.