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------------------------------------------------ At Company Z, machines that fill bottles with water are supposed to dispense 16 ounces into each bottle. Let [tex]X[/tex] be the weight (in ounces) of the water dispensed into a randomly selected bottle. Suppose that [tex]X[/tex] can be modeled by a Normal distribution with a mean of 16 ounces and a standard deviation of 0.5 ounces.

Which of the following is a correct interpretation of the value of [tex]P(X \leq 15)[/tex]?

A. [tex]P(X \leq 15) = 0.25[/tex]. About 25% of bottles dispensed at Company Z weigh less than 15 ounces.

B. [tex]P(X \leq 15) = 0.977[/tex]. About 97.7% of water bottles from all companies' machines are dispensed with 15 ounces or less.

C. [tex]P(X \leq 15) = 0.023[/tex]. About 2.3% of water bottles from all companies' machines are dispensed with 15 ounces or less.

D. [tex]P(X \leq 15) = 0.023[/tex]. About 2.3% of water bottles from Company Z's machines are dispensed with 15 ounces or less.

E. [tex]P(X \leq 15) = 0.977[/tex]. About 97.7% of water bottles from Company Z's machines are dispensed with 15 ounces or less.

To solve this problem, we need to interpret the probability [tex]$P(X \leq 15)$[/tex], where [tex]$X$[/tex] is a normally distributed random variable representing the weight of water dispensed into a bottle. The mean ([tex]$\mu$[/tex]) of this distribution is 16 ounces, and the standard deviation ([tex]$\sigma$[/tex]) is 0.5 ounces.

1. Calculate the z-score:
The z-score tells us how many standard deviations an element is from the mean. The formula to calculate the z-score for a value [tex]$X$[/tex] is:

[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]

Substituting the given values:

[tex]\[
z = \frac{15 - 16}{0.5} = -2.0
\][/tex]

This z-score of -2.0 indicates that 15 ounces is 2 standard deviations below the mean.

2. Find the probability associated with the z-score:
We need to find [tex]$P(X \leq 15)$[/tex], which is the cumulative probability up to the z-score of -2.0. You can look up this value in the standard normal distribution table (z-table) or use a statistical calculator.

For a z-score of -2.0, the cumulative probability [tex]$P(Z \leq -2.0)$[/tex] is approximately 0.0228, often rounded to 0.023.

3. Interpret the probability:
This probability, [tex]$P(X \leq 15) = 0.023$[/tex], means that about 2.3% of the bottles dispensed by Company Z's machines contain 15 ounces or less of water.

Thus, the correct interpretation of the given options is:
- [tex]$P(X \leq 15)=0.023$[/tex]. About 2.3% of water bottles from Company Z's machines are dispensed with 15 ounces or less.