Question

assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of \\( \mu = 1.1 \mathrm { kg } \\) and a standard deviation of \\( \sigma = 4.5 \mathrm { kg } \\). complete parts (a) through (c) below.
a. if 1 male college student is randomly selected, find the probability that he gains between 0 kg and 3 kg during freshman year.
the probability is \\( \square \\).
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
For \(x = 0\), \(z_1=\frac{0 - 1.1}{4.5}=\frac{-1.1}{4.5}\approx - 0.24\)
For \(x = 3\), \(z_2=\frac{3 - 1.1}{4.5}=\frac{1.9}{4.5}\approx0.42\)

Step2: Find the probabilities corresponding to the z - scores

Using the standard normal distribution table (or a calculator with a normal - distribution function), we know that \(P(Z\lt - 0.24)\) and \(P(Z\lt0.42)\)
\(P(Z\lt - 0.24)=0.4052\) (from the standard normal table)
\(P(Z\lt0.42) = 0.6628\) (from the standard normal table)

Step3: Calculate the probability between the two z - scores

The probability \(P(-0.24\lt Z\lt0.42)=P(Z\lt0.42)-P(Z\lt - 0.24)\)
\(P(-0.24\lt Z\lt0.42)=0.6628 - 0.4052=0.2576\)

Answer:

\(0.2576\)