i don't really know how to go about anwsering this problem:
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?
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.
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.
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.
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.
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.
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
To solve this problem, we need to use the properties of a normal distribution. Given that the weight of the eggs produced by the breed of hen is normally distributed with a mean of 65 grams and a standard deviation of 5 grams, we can calculate the probability that the weight of a carton falls between 750g and 825g.
To achieve this, we will need to transform the given information into a standard normal distribution with a mean of 0 and a standard deviation of 1.
Step 1: Calculate the sample mean and standard deviation for the carton.
Since a carton contains 12 eggs, the mean weight of a carton is 12 times the mean weight of a single egg, which is 12 * 65 = 780 grams.
Similarly, the standard deviation of a carton is the square root of the sum of the variances of the 12 eggs, which is √(12 * (5)^2) = √(12 * 25) = √300 = 5√6 grams.
Step 2: Standardize the values.
To standardize the values in order to convert them to a standard normal distribution, we need to subtract the sample mean from 750g and 825g, respectively, and then divide the differences by the sample standard deviation (5√6).
Standardizing 750g:
Z1 = (750 - 780) / (5√6)
Standardizing 825g:
Z2 = (825 - 780) / (5√6)
Step 3: Calculate the probabilities using the standard normal distribution.
Using a standard normal distribution table or a calculator, we can find the probabilities associated with the standardized values Z1 and Z2.
P(750g < weight of carton < 825g) = P(Z1 < Z < Z2)
Here, Z represents the standard score (or Z-score).
Step 4: Look up the probabilities.
Using the standard normal distribution table or a calculator, find the probabilities P(Z < Z1) and P(Z < Z2), then find P(Z1 < Z < Z2) by subtracting P(Z < Z1) from P(Z < Z2).
P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
This will give you the probability that the weight of a carton falls between 750g and 825g.
Please note that if you are using a calculator, you can directly input the values of Z1 and Z2 to get the probability without going through the standard normal distribution table.