Question

Mean daily milk production for Jersey cows is 43 pounds with a known population standard deviation of 5 pounds and we know that milk production for these cows follows a Normal model. Suppose a farmer has a small herd of 20 Jersey cows. How do we know that the distribution of the sample mean, , for the small herd of cows, is approximately Normal?

Answer 1

Solution:

Given:

X = The milk production for Jersey cows follows a Normal model with Mean daily milk production is 43 pounds with a known population standard deviation of 5 pounds.

Sample size = n= 20

Since population of the milk production for Jersey cows follows a Normal model, then sampling distribution of sample mean is also follows a Normal distribution, regardless of sample size n.

We need large sample size n ( at least 30) to assume sampling distribution of sample mean approximately Normal, but in case of Normal distribution, it is not necessary of large sample.

Thus  the distribution of the sample mean is Normal with mean of sample means:

and

standard deviation of sample means is: