Answer :
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.
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.
