Question

independent practice
the mean weight of adult males is 172 pounds with standard deviation 43 pounds. calculate the probability that the mean weight of 25 randomly selected males is less than 190 pounds.
98.183%
66.225%
54.417%
47.379%

Explanation:

Step1: Calculate the standard error

The standard error ($SE$) is given by the formula $SE=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
Here, $\sigma = 43$ and $n = 25$.
So, $SE=\frac{43}{\sqrt{25}}=\frac{43}{5} = 8.6$

Step2: Calculate the z - score

The z - score is calculated using the formula $z=\frac{\bar{x}-\mu}{SE}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean.
Here, $\bar{x}=190$, $\mu = 172$, and $SE = 8.6$
So, $z=\frac{190 - 172}{8.6}=\frac{18}{8.6}\approx2.093$

Step3: Find the probability

We want to find $P(\bar{X}<190)$. Using the standard normal distribution table (or a calculator with a normal - distribution function), for $z = 2.093$, $P(Z<2.093)\approx0.98183$ or $98.183\%$

Answer:

98.183%