Sure, let's go through this step-by-step to determine the percentage of onions that will be accepted by the grocery store.
1. Understanding the Distribution:
We know that the diameters of the onions follow a normal distribution with:
- Mean (μ) = 2.75 inches
- Standard Deviation (σ) = 0.25 inches
2. Defining the Acceptance Criteria:
The grocery store accepts onions with a diameter of at least 3.25 inches.
3. Calculating the Z-Score:
To find out how far 3.25 inches is from the mean in terms of standard deviations, we calculate the Z-score. The formula for the Z-score is given by:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
Where:
- [tex]\( X \)[/tex] is the value we are interested in (3.25 inches in this case)
- [tex]\( \mu \)[/tex] is the mean (2.75 inches)
- [tex]\( \sigma \)[/tex] is the standard deviation (0.25 inches)
Plugging in the values, we get:
[tex]\[
Z = \frac{3.25 - 2.75}{0.25} = \frac{0.50}{0.25} = 2.0
\][/tex]
So, the Z-score is 2.0.
4. Finding the Cumulative Probability:
The Z-score tells us how many standard deviations away 3.25 inches is from the mean. To find the probability that an onion will be less than 3.25 inches, we look up the cumulative distribution function (CDF) for Z = 2.0.
The CDF value for Z = 2.0 is approximately 0.9772. This value represents the probability that a randomly selected onion has a diameter less than 3.25 inches.
5. Calculating the Acceptance Probability:
Since we want the probability of onions having a diameter of at least 3.25 inches, we need to find the complement of the cumulative probability:
[tex]\[
P(X \geq 3.25) = 1 - P(X < 3.25) = 1 - 0.9772 = 0.0228
\][/tex]
Hence, the probability of an onion being accepted is approximately 0.0228.
6. Converting to Percentage:
To find the percentage of onions that will be accepted by the grocery store, we convert the probability into a percentage:
[tex]\[
\text{Percentage Accepted} = 0.0228 \times 100 = 2.28\%
\][/tex]
Therefore, approximately [tex]\(2.28\% \)[/tex] of the onions will be accepted by the grocery store.
