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) kg and a standard deviation of (sigma = 4.5) 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.2576
(round to four decimal places as needed.)

b. if 4 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg

the probability is (square)
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the standard deviation of the sample mean

For a sample of size \(n = 4\), the standard deviation of the sample mean \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 4.5\) kg and \(n = 4\), we have \(\sigma_{\bar{x}}=\frac{4.5}{\sqrt{4}}=\frac{4.5}{2}=2.25\) kg.

Step2: Calculate the z - scores

The z - score formula is \(z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}\).
For \(\bar{x}=0\): \(z_1=\frac{0 - 1.1}{2.25}=\frac{- 1.1}{2.25}\approx - 0.49\)
For \(\bar{x}=3\): \(z_2=\frac{3 - 1.1}{2.25}=\frac{1.9}{2.25}\approx0.84\)

Step3: Find the probabilities using the standard normal distribution table

\(P(-0.49<Z<0.84)=P(Z < 0.84)-P(Z<-0.49)\)
From the standard normal distribution table, \(P(Z < 0.84)=0.7995\) and \(P(Z<-0.49) = 0.3121\)
\(P(-0.49<Z<0.84)=0.7995 - 0.3121=0.4874\)

Answer:

\(0.4874\)