Question

the weights of college football players are normally distributed with a mean of 200 pounds and a standard deviation of 50 pounds. if a college football player is randomly selected, find the probability that he weighs between 170 and 220 pounds. round to four decimal places.
a. 0.3812
b. 0.2257
c. 0.1554
d. 0.0703

Explanation:

Step1: Calculate the z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 200\), \(\sigma=50\).
For \(x = 170\): \(z_1=\frac{170 - 200}{50}=\frac{- 30}{50}=-0.6\)
For \(x = 220\): \(z_2=\frac{220 - 200}{50}=\frac{20}{50}=0.4\)

Step2: Use the standard normal distribution table

We want to find \(P(-0.6<Z<0.4)\).
Since \(P(-0.6 < Z < 0.4)=P(Z < 0.4)-P(Z<-0.6)\)
From the standard normal table, \(P(Z < 0.4)=0.6554\) and \(P(Z<-0.6)=0.2743\)

Step3: Calculate the probability

\(P(-0.6 < Z < 0.4)=0.6554-0.2743 = 0.3811\approx0.3812\)

Answer:

A. 0.3812