Difference between Z-test and t-test

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Blue7195
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#1
I'm stuck with how to tell the difference between a Z-test and t-test. I always just thought Z-test was normal distribution and t-test was t-distribution, Z test was for >=30 and t test was for < 30. However, I have this example which ends is a two-sided t-test in the answers. I am just not too sure why and would love some feedback on why it is?

The researcher found that 12 of the 31 women had taken the oral contraceptive pill at some time and that 19 had not. He obtained the following results relating to age at menopause, and concluded that taking the pill does not delay menopause:

Pill Users: n=12, mean=47.2, SD=2.1
Non Pill Users: n=19, mean = 47.8, SD=2.1

All i can think of is since n=12 and n=19 - however I thought I would look at the sample size of n=31 and deduce the statistical test from that?
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