Probability Question - Nonstandard Normal Distributions

[Love_to_Learn]

Homework Statement

The weight of eggs produced by a certain type of hen varies according to a distribution that is approximately normal with mean 6.5 grams and standard deviation 2 grams.

What is the probability that the average of a random sample of the weights of 25 eggs will be less than 6 grams

Homework Equations

P(X<6)=P((6-6.5)/σ)

The Attempt at a Solution

- The part I can't figure out is how to arrive at sigma. This is a problem from a practice exam, so I already know that sigma is 0.40. If I'm understanding correctly, then that would make V(X) = 4/25. I just can't figure out how to arrive at these conclusion from the data that is given. I'm pretty stumped.

 

[statdad]

your problem asks about the probability the MEAN of a sample will be a certain size. what do you know about distributions of sample means?

 

[Love_to_Learn]

your problem asks about the probability the MEAN of a sample will be a certain size. what do you know about distributions of sample means?

Not much.

I think I figured out why sigma is what it is though. Since the SD is 2 grams, it follows that V(X) = 4. Since I'm trying to find out what the average of X is, I divide 4 by 25. that is how I get the 4/25. From there sigma is easy. I think that is kind of close anyway.

 

[statdad]

You've essentially got it. If you take a sample of size $$n$$ from a normally distributed population, then $$\overline X$$ has a normal distribution. For the
distribution of $$\overline X$$,

$$\mu = \text{ original population mean}$$

and

$$\sigma = \frac{\text{Original standard deviation}}{\sqrt n}$$

As long as the original population itself has a normal distribution, this is true
for any sample size.