Question

machines that fill bags with powdered sugar are supposed to dispense 32 ounces of powdered sugar into each bag. let ( x = ) the weight (in ounces) of the powdered sugar dispensed into a randomly selected bag. the probability distribution of ( x ) is approximately normal, with mean 32 ounces and standard deviation 0.6 ounce.
find ( p(x leq 31) ).
(round 4 decimal places. leave your answer in decimal form.)
describe this probability in words.
there is about a (\boxed{}) probability that a (\boxed{}) will weigh less than or equal to 31 ounces.

Explanation:

Step1: Identify the distribution and parameters

We have a normal distribution with mean $\mu = 32$ ounces and standard deviation $\sigma = 0.6$ ounce. We need to find $P(X \leq 31)$. First, we calculate the z - score using the formula $z=\frac{x-\mu}{\sigma}$.
Substitute $x = 31$, $\mu = 32$, and $\sigma=0.6$ into the z - score formula: $z=\frac{31 - 32}{0.6}=\frac{- 1}{0.6}\approx - 1.67$ (rounded to two decimal places).

Step2: Find the probability from the z - table

We need to find $P(Z\leq - 1.67)$ where $Z$ is the standard normal variable. Looking up the z - score of - 1.67 in the standard normal distribution table (or using a calculator with a normal distribution function), we know that the cumulative probability for $z=-1.67$ is approximately 0.0475.

Answer:

0.0475