Answer:
About 34% of the weights are between 3280 grams and 3720 grams.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean of 3500 grams, standard deviation of 500 grams.
This means that ![\mu = 3500, \sigma = 500](https://tex.z-dn.net/?f=%5Cmu%20%3D%203500%2C%20%5Csigma%20%3D%20500)
About 34% of the weights are between what two values?
Between the 50 - (34/2) = 33rd percentile and the 50 + (34/2) = 67th percentile.
33rd percentile:
X when Z has a p-value of 0.33, so X when Z = -0.44.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-0.44 = \frac{X - 3500}{500}](https://tex.z-dn.net/?f=-0.44%20%3D%20%5Cfrac%7BX%20-%203500%7D%7B500%7D)
![X - 3500 = -0.44*500](https://tex.z-dn.net/?f=X%20-%203500%20%3D%20-0.44%2A500)
![X = 3280](https://tex.z-dn.net/?f=X%20%3D%203280)
67th percentile:
X when Z has a p-value of 0.67, so X when Z = 0.44.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![0.44 = \frac{X - 3500}{500}](https://tex.z-dn.net/?f=0.44%20%3D%20%5Cfrac%7BX%20-%203500%7D%7B500%7D)
![X - 3500 = 0.44*500](https://tex.z-dn.net/?f=X%20-%203500%20%3D%200.44%2A500)
![X = 3720](https://tex.z-dn.net/?f=X%20%3D%203720)
About 34% of the weights are between 3280 grams and 3720 grams.