The diameters of apples in a certain orchard are normally distributed with a mean of 4.77 inches and a standard deviation of 0.43 inches. Show all work.

(A) What percentage of the apples in this orchard is larger than 4.71 inches?
(B) A random sample of 100 apples is gathered and the mean diameter is calculated. What is the probability that the sample mean is greater than 4.71 inches?

To solve these questions, we need to standardize the value using the z-score formula and then use the standard normal distribution table or a calculator that provides the cumulative distribution function (CDF) for the standard normal distribution.

The z-score formula is given by:
z = (X - μ) / σ

Where:
X is the value we want to standardize (the diameter in this case),
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.

(A) To find the percentage of apples larger than 4.71 inches, we need to calculate the z-score for this value and then find the area under the normal curve to the right of this z-score.
z = (4.71 - 4.77) / 0.43
z = -0.06 / 0.43
z ≈ -0.14

Using a standard normal distribution table or a calculator, we can find that the area (or probability) to the right of z = -0.14 is approximately 0.556 (or 55.6%).

Therefore, the percentage of apples in this orchard that is larger than 4.71 inches is approximately 55.6%.

(B) To find the probability that the sample mean of 100 apples is greater than 4.71 inches, we first need to calculate the standard error of the mean (SEM).
SEM = σ / sqrt(n)

Where:
σ is the standard deviation of the population (0.43 inches in this case), and
n is the sample size (100 apples in this case).

SEM = 0.43 / sqrt(100)
SEM = 0.43 / 10
SEM = 0.043

Now, we can calculate the z-score for the sample mean using the z-score formula:
z = (X - μ) / SEM

In this case, X is 4.71 inches, μ is still 4.77 inches, and SEM is 0.043 inches.

z = (4.71 - 4.77) / 0.043
z = -0.06 / 0.043
z ≈ -1.395

Using a standard normal distribution table or a calculator, we can find the area (or probability) to the right of z = -1.395 is approximately 0.082 (or 8.2%).

Therefore, the probability that the sample mean of 100 apples is greater than 4.71 inches is approximately 8.2%.