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T distribution

When we use T distribution?

When we know, population is normal distribution and what we have is only sample.

What is the question to which T distribution answers?

Given a small sample out of population with an assumed/declared
population mean, what is the cumulative probability that another trail that I conduct by taking same sample size of previous sample trail, results in same as earlier sample mean?

Example: I say my population mean is 10 defective bulbs out of 1000.
Let us conduct a test on 30 random samples. Let us say trail shows 12
defective bulbs out of 30 sample size. If I do What is the probability that in next trial we can again have 12 defective bulbs?



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