Question 1184133: The average weight of incoming freshmen of the PMA is 150lbs. with a standard deviation of 15lbs. (Assume normal distribution for the weight).
a. What is the probability of the incoming freshmen weighing between 125 and 175 lbs.?
b. What is the minimum weight of the heftiest 10% of the incoming freshmen?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**a. Probability of Weight between 125 and 175 lbs:**
1. **Calculate the z-scores:**
A z-score tells you how many standard deviations a value is from the mean. The formula is:
z = (x - μ) / σ
Where:
* x is the value you're interested in
* μ is the mean (150 lbs)
* σ is the standard deviation (15 lbs)
* For x = 125 lbs: z1 = (125 - 150) / 15 = -1.67
* For x = 175 lbs: z2 = (175 - 150) / 15 = 1.67
2. **Look up the probabilities in a z-table or use a calculator:**
A z-table (or a statistical calculator) gives you the probability of a value being *less than* a given z-score.
* Find the probability associated with z = 1.67. This gives you the area to the left of 175lbs. A z-table will show a value of 0.9525.
* Find the probability associated with z = -1.67. This gives you the area to the left of 125lbs. A z-table will show a value of 0.0475.
3. **Calculate the probability between the two values:**
Subtract the smaller probability from the larger probability: 0.9525 - 0.0475 = 0.905
Therefore, the probability of an incoming freshman weighing between 125 and 175 lbs is approximately 0.905 or 90.5%.
**b. Minimum Weight of the Heftiest 10%:**
1. **Find the z-score corresponding to the top 10%:**
Since we're looking for the *minimum* weight of the *heaviest* 10%, we want the weight that separates the top 10% from the rest. This means we are looking for the 90th percentile. So, we are looking for the z-score where 90% of the data lies below it. Look up 0.90 in the *body* of the z-table (or use a calculator's inverse normal function) to find the corresponding z-score. It is approximately 1.28.
2. **Use the z-score formula to find the weight:**
We know the z-score (1.28), the mean (150), and the standard deviation (15). We want to find x:
z = (x - μ) / σ
1.28 = (x - 150) / 15
Solve for x:
x = (1.28 * 15) + 150
x = 19.2 + 150
x = 169.2 lbs
Therefore, the minimum weight of the heftiest 10% of the incoming freshmen is approximately 169.2 lbs.
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