Question

the weight of oranges growing in an orchard is normally distributed with a mean weight of 3.5 oz. and a standard deviation of 1 oz. what is the probability that a randomly selected orange from the orchard weighs less than 2 oz., to the nearest thousandth? statistics calculator

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $x$ is the value from the dataset, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Given that $\mu = 3.5$, $\sigma=1$, and $x = 2$.
Substitute these values into the z - score formula:
$z=\frac{2 - 3.5}{1}=\frac{- 1.5}{1}=-1.5$

Step2: Find the probability using the z - score

We need to find $P(Z < - 1.5)$ where $Z$ follows the standard normal distribution.
Using a standard normal table (z - table) or a calculator with a normal distribution function, we know that the cumulative probability for $z=-1.5$ can be found.
Looking up $z = - 1.5$ in the standard normal table, we find that $P(Z < - 1.5)=0.0668\approx0.067$ (to the nearest thousandth)

Answer:

$0.067$