i don't really know how to go about anwsering this problem:

Question

the weight of the eggs produced by a certain breed of hen is Normally distributed with mean 65 grams and standard deviation 5 grams. think of cartons of such eggs as SRSs of size 12 from the population of all eggs.
what is the probabiltythat the weight of a carton falls between 750g and 825g?

any help?

Answer 1

I think they are saying that there are 12 eggs in a carton
So the mean of a carton is 780 g with a standard deviation of 24 g.

We need the z-score for both 825 and 750

z-score for 825 = (825-780)/24 = 1.875
z-score for 750 = (750-780)/24 = - 1.25

Now go to your Normal Distribution Table ( I don't have one handy)
look up the value for 1.875 and subtract from it the value for -1.25

That decimal value will be your probability.

Answer 2

The mean weight of 12 eggs will be 12x65 = 780 g and the standard deviation will be sqrt(120) times the stadard deviation for a single egg, or 17.32 g. Integrate the normal distribution between those limits.

Using this helpful website:
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html ,

I get 95.4% probability that the weight is between the specified limits.

Answer 3

I am not in agreement with Reiny's statement that the standard deviation of the weight of 12 eggs is 24. In my previous answer, sqrt(120) should have been sqrt(12), and my assumption for the standard deviation was based on that.

Answer 4

You are so right drwls, I don't know how I possible got 5 x 12 to be 24, it should have been 60

so let me try again:
I think they are saying that there are 12 eggs in a carton
So the mean of a carton is 780 g with a standard deviation of 60 g.

We need the z-score for both 825 and 750

z-score for 825 = (825-780)/60 = 0.75
z-score for 750 = (750-780)/60 = - 0.5

Now go to your Normal Distribution Table ( I don't have one handy)
look up the value for 0.75 and subtract from it the value for -0.5

That decimal value will be your probability.

Answer 5

I still disagree. The standard deviation of the sum of N measurements with a normal distribution does not scale with N. As I recall, it scales with the square root of N. That is why I multiplied 5 by the square root of 12.

My knowledge of probability theory is a bit rusty, so I could be wrong.

Answer 6

The square root enters the picture as we calculate standard deviation.
But unless I am missing something here, the standard deviation is given, thus already calculated.

Suppose I change the question a bit and look only at one egg instead of one carton. Then it would read

"the weight of the eggs produced by a certain breed of hen is Normally distributed with mean 65 grams and standard deviation 5 grams.
what is the probabiltythat the weight of an egg falls between 62.5g and 68.75g?" (divided the carton weight by 12)

Using
http://davidmlane.com/hyperstat/z_table.html
I obtain the same answer as obtained from my above calculation