Question

Suppose you had a population of fish whose lengths were normally distributed with a mean of 50 centimeters and a standard deviation of 5 centimeters. You draw a simple random sample of size 10, record the length of each fish, and calculate the mean of the sample lengths. What do you think your sample mean would most likely be

Answer 1

The sample mean would most likely be 50 centimeters.

Explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

In this question:

The mean of the population is 50 centimers.

So, by the Central Limit Theorem, the sample mean would most likely be also 50 centimeters.