To determine how many of the 500 soft drinks served in a day would be expected to have at least 8.5 ounces, we need to analyze the distribution of the ounces of the soft drinks.
1. Assumptions:
- The ounces of the soft drinks served follow a normal distribution.
- The mean (average) amount of each soft drink is 8 ounces.
- The standard deviation (a measure of the spread) of the ounces is 2 ounces.
2. Calculate the z-score:
- The z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Where:
- [tex]\(X\)[/tex] is the value in question (8.5 ounces in this case),
- [tex]\(\mu\)[/tex] is the mean (8 ounces),
- [tex]\(\sigma\)[/tex] is the standard deviation (2 ounces).
Substituting the values, we get:
[tex]\[
z = \frac{8.5 - 8}{2} = \frac{0.5}{2} = 0.25
\][/tex]
3. Find the probability corresponding to the z-score:
- To find the probability that a value is at least 8.5 ounces, we use the cumulative distribution function (CDF) of the normal distribution.
- The CDF gives the probability that a value is less than or equal to a given z-score.
For a z-score of 0.25, the CDF value is approximately 0.5987. This represents the probability of a soft drink being less than 8.5 ounces.
- To find the probability of a soft drink being at least 8.5 ounces, we subtract this value from 1:
[tex]\[
P(X \geq 8.5) = 1 - P(X < 8.5) = 1 - 0.5987 = 0.4013
\][/tex]
4. Calculate the expected number of soft drinks:
- We now multiply the total number of soft drinks by this probability to get the expected number of soft drinks that are at least 8.5 ounces:
[tex]\[
\text{Expected number of soft drinks} = 500 \times 0.4013 = 200.65
\][/tex]
Therefore, out of 500 soft drinks served, we would expect approximately 200.65, rounded to 200, to have at least 8.5 ounces. This means none of the provided options (24, 28, 26, 22) correctly represent the expected number of soft drinks that are at least 8.5 ounces, based on the given distribution.
