Relationship between sample size, population mean and std deviation?

It would be a help to know the population distribution but as we don’t know any better let’s assume it’s a Normal Distribution.

The sample means will be normally distributed with mean 30 and sd = 3/√5 = 1.34.

Mean of 35 has a z statistic z=5/sd = 3.73. From the Normal Distribution tables probability z>3.73 is 0.0001.

If we take ten samples then the number with mean >3.5 will be a Binomial Distribution with n=10 p=0.0001. The mean of this distribution is np=0.001.

You take ten samples from the population. You ask of each sample the question - is the mean greater than 35? Then there is a new distribution (with integer values) which is the number of times (out of 10) that the answer is 'yes'. That's a Binomial distribution.

Lets think a bit about Binomial distributions. Suppose you throw a normal die nine times then the number of sixes thrown will be a Binomial distribution n=9 p=1/6. How many sixes would you expect to see? Well in normal language you would say one or two! Actually probability of one is 0.35 and probability of two is 0.30. Other possibilities are relatively small.

In mathematical language the 'expected value' is the mean of the distribution in this case 1.5 [np from the Binomial distribution parameters]. And although each possible value of the distribution is an integer this 'expected value' is not.

In your case the event is so unlikely that in normal language we would say the expected number of samples with mean over 35 is zero; we don't actually expect to see any at all. Still I take the word 'expect' to refer to the mathematical term 'expected value' then i answer 0.001 as above.

4. mean 50 and standard error 2