the diameter of red apples of an orchard have a normal distribution with a mean of 3 inches ans standard deviation of 0.5 inches one apple will be randomly chosen. what is the probability of picking an apple with a diameter less than 2.15 inches

2.15 is .85 below the mean

this is .85/.5 s.d. (1.7)

use a z-score table to find the fraction of the population in this range

about .045

To solve this problem, we will use the information provided about the diameter of red apples in the orchard. We are given that the diameter follows a normal distribution with a mean of 3 inches and a standard deviation of 0.5 inches.

To find the probability of picking an apple with a diameter less than 2.15 inches, we need to calculate the area to the left of this value on the normal distribution curve. This can be done using the standard normal distribution table or by using statistical software.

Using the standard normal distribution table, we convert the value of 2.15 inches to a z-score. The z-score represents the number of standard deviations the diameter of 2.15 inches is from the mean of 3 inches.

To calculate the z-score, we use the formula:

z = (x - μ) / σ

Where:
- x is the value we want to convert to a z-score (2.15 inches).
- μ is the mean of the distribution (3 inches).
- σ is the standard deviation of the distribution (0.5 inches).

Plugging the values into the formula:

z = (2.15 - 3) / 0.5
z = -0.7

The z-score of -0.7 indicates that the diameter of 2.15 inches is 0.7 standard deviations below the mean.

Now, we can refer to the standard normal distribution table to find the corresponding cumulative probability for a z-score of -0.7. The cumulative probability represents the area under the curve to the left of the z-score.

Consulting the table, we find that the cumulative probability for a z-score of -0.7 is approximately 0.2420.

Therefore, the probability of picking an apple with a diameter less than 2.15 inches is approximately 0.2420, or 24.20%.