The USDA size standards for eggs are based on weight and are as follows:
SMALL: greater than 43 grams, less than 50 grams
MEDIUM: greater than 50 grams, less than 57 grams
LARGE: greater than 57 grams, less than 64 grams
EXTRA LARGE: greater than 64 grams, less than 71 grams
JUMBO: greater than 71 grams.
The output of a particular egg farm has a mean weight of 62 grams with a standard deviation of 8 grams. What is the percentage of egg output of this farm in each size category?
Z = (score - mean)/standard deviation
Small eggs are between 43-50 grams.
(43-62)/8 ≤ Small eggs ≤ (50-62)/8
Look up these two z scores in table in back of your stat text labeled something like "areas under normal distribution" to find proportion between these two points. Convert to percentage.
Do the same with the other categories.
I hope this helps.
To determine the percentage of egg output in each size category, we need to calculate the proportion of eggs that fall within each weight range.
Let's start by calculating the Z-scores for each size category using the formula:
Z-score = (x - mean) / standard deviation
Where:
x = the weight limit for the size category
For the Small category, the Z-score would be:
Z-small = (50 - 62) / 8
For the Medium category, the Z-score would be:
Z-medium = (57 - 62) / 8
For the Large category, the Z-score would be:
Z-large = (64 - 62) / 8
For the Extra Large category, the Z-score would be:
Z-extra_large = (71 - 62) / 8
Next, we will use the Z-scores to calculate the proportions using a standard normal distribution table or a Z-score calculator.
The proportion for the Small category would be:
P-small = P(Z ≤ Z-small)
The proportion for the Medium category would be:
P-medium = P(Z-small< Z ≤ Z-medium)
The proportion for the Large category would be:
P-large = P(Z-medium < Z ≤ Z-large)
The proportion for the Extra Large category would be:
P-extra_large = P(Z-large < Z ≤ Z-extra_large)
Finally, to determine the percentage of egg output in each size category, multiply the proportions by 100.
Percentage for Small category = P-small * 100
Percentage for Medium category = P-medium * 100
Percentage for Large category = P-large * 100
Percentage for Extra Large category = P-extra_large * 100
It's important to note that these calculations assume a normal distribution of egg weights.