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?

A. Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.

B. Same process except that, instead of SD, you use SEm = SD/√n

once again I am not getting this.

assume that the population of heights of male college students is normally distibuted with a mean of 69.09 and standard deviation of 4.71. A random sample of 92 heights is obtained. find the mean and standard error of the x distribution.
find P(x>68.5)

To find the answer to these questions, we will use the standard normal distribution. Given the mean and standard deviation of the population, we can convert any value to its z-score and then find the corresponding area under the standard normal curve using a z-table or a statistical calculator.

(A) To find the percentage of apples larger than 4.71 inches, we need to determine the area to the right of 4.71 on the normal curve.

Step 1: Calculate the z-score
The z-score formula is:
z = (x - μ) / σ

where x is the value of interest (4.71 inches), μ is the mean (4.77 inches), and σ is the standard deviation (0.43 inches).

z = (4.71 - 4.77) / 0.43 = -0.06 / 0.43 ≈ -0.1395

Step 2: Find the area to the right of the z-score
Since we're looking for the percentage of apples larger than 4.71 inches, we need to find the area to the right of the z-score (-0.1395).

Using a z-table or a statistical calculator, we find that the area to the right of -0.1395 is approximately 0.5564.

Step 3: Calculate the percentage
To convert the area to a percentage, we multiply it by 100:

Percentage = 0.5564 × 100 ≈ 55.64%

Therefore, approximately 55.64% of the apples in this orchard are larger than 4.71 inches.

(B) To find the probability that the sample mean is greater than 4.71 inches for a random sample of 100 apples, we use the central limit theorem. According to the central limit theorem, if the sample size is sufficiently large (typically above 30) and the population is normally distributed, the distribution of the sample means will be approximately normal, regardless of the shape of the population.

In this case, the mean of the sample means will still be 4.77 inches (the same as the population mean), but the standard deviation of the sample means will be the population standard deviation divided by the square root of the sample size (√n).

Step 1: Calculate the standard deviation of the sample means
standard deviation of the sample means = σ / √n

For n = 100, the standard deviation of the sample means is 0.43 / √100 = 0.43 / 10 = 0.043

Step 2: Calculate the z-score for the sample mean
Using the same formula as before:
z = (x - μ) / (σ / √n)

z = (4.71 - 4.77) / 0.043 ≈ -1.3953

Step 3: Find the area to the right of the z-score
Using a z-table or a statistical calculator, we find that the area to the right of -1.3953 is approximately 0.0823.

Therefore, the probability that the sample mean is greater than 4.71 inches is approximately 0.0823, or 8.23%.