A random sample of size 36 is to be selected from a population that has a mean μ = 50 and a standard deviation σ of 10.

* a. This sample of 36 has a mean value of , which belongs to a sampling distribution. Find the shape of this sampling distribution.
* b. Find the mean of this sampling distribution.
* c. Find the standard error of this sampling distribution.
* d. What is the probability that this sample mean will be between 45 and 55?
* e. What is the probability that the sample mean will have a value greater than 48?
* f. What is the probability that the sample mean will be within 3 units of the mean?

a. The shape of the sampling distribution can be assumed to be approximately normal due to the Central Limit Theorem, which states that the distribution of sample means becomes increasingly normal as the sample size increases, regardless of the shape of the population distribution.

b. The mean of the sampling distribution is equal to the mean of the population, which is μ = 50.

c. The standard error (SE) of the sampling distribution is equal to the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation of the population is σ = 10, and the sample size is 36. Therefore, the standard error is:

SE = σ / √n
SE = 10 / √36
SE = 10 / 6
SE = 1.67

d. To find the probability that the sample mean will be between 45 and 55, we need to convert the sample mean values to z-scores and then use the z-table. The formula to calculate the z-score is:

z = (x - μ) / SE

Using this formula, we can calculate the z-scores for 45 and 55:

For x = 45:
z1 = (45 - 50) / 1.67
z1 ≈ -3.00

For x = 55:
z2 = (55 - 50) / 1.67
z2 ≈ 3.00

Using the z-table, we find the probability corresponding to a z-score of -3.00 is approximately 0.0013. Since the distribution is symmetric, the probability corresponding to a z-score of 3.00 will be the same. Therefore, the probability that the sample mean will be between 45 and 55 is approximately 2 * 0.0013 = 0.0026.

e. To find the probability that the sample mean will have a value greater than 48, we need to calculate the z-score for 48 and then use the z-table. Using the formula:

z = (x - μ) / SE

For x = 48:
z = (48 - 50) / 1.67
z ≈ -1.20

Using the z-table, we find the probability corresponding to a z-score of -1.20 is approximately 0.1151. Since we want the probability of the sample mean being greater than 48, we need to subtract this probability from 1:

P(sample mean > 48) = 1 - 0.1151 = 0.8849

Therefore, the probability that the sample mean will have a value greater than 48 is approximately 0.8849.

f. To find the probability that the sample mean will be within 3 units of the mean, we need to calculate the z-scores for the values 50 - 3 and 50 + 3, and then use the z-table.

For x = 50 - 3 = 47:
z1 = (47 - 50) / 1.67
z1 ≈ -1.80

For x = 50 + 3 = 53:
z2 = (53 - 50) / 1.67
z2 ≈ 1.80

Using the z-table, we find the probability corresponding to a z-score of -1.80 is approximately 0.0359. Since the distribution is symmetric, the probability corresponding to a z-score of 1.80 will be the same. Therefore, the probability that the sample mean will be within 3 units of the mean is approximately 2 * 0.0359 = 0.0718.

Therefore, the probability that the sample mean will be within 3 units of the mean is approximately 0.0718.

To answer these questions, we need to understand the concept of sampling distributions. A sampling distribution is the probability distribution of a statistic (in this case, the sample mean) based on a random sample from a population.

Let's calculate the answers step by step:

a. The shape of the sampling distribution is approximately normal. This is because according to the central limit theorem, for large sample sizes (n > 30), the sampling distribution of the sample mean will be approximately normal regardless of the shape of the population.

b. The mean of the sampling distribution is equal to the population mean, μ. Therefore, the mean of this sampling distribution is 50.

c. The standard error (SE) of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size (√n). Therefore, the standard error of this sampling distribution is σ/√n = 10/√36 = 10/6 = 1.67.

d. To find the probability that the sample mean will be between 45 and 55, we can use the z-score formula and consult the standard normal distribution table or use statistical software. The z-score is calculated as (X - μ) / SE, where X is the sample mean, μ is the population mean, and SE is the standard error.

For X = 45:
z-score = (45 - 50) / 1.67 = -3.01 (approximately)

For X = 55:
z-score = (55 - 50) / 1.67 = 3.01 (approximately)

Using the standard normal distribution table or software, we can find the area under the curve between -3.01 and 3.01, which represents the probability. It is approximately 0.997 (or 99.7%).

e. To find the probability that the sample mean will have a value greater than 48, we need to calculate the z-score for X = 48 and find the area under the curve to the right of that z-score.

z-score = (48 - 50) / 1.67 = -1.20 (approximately)

Using the standard normal distribution table or software, we can find the area to the right of -1.20, which is approximately 0.885 (or 88.5%).

f. To find the probability that the sample mean will be within 3 units of the mean, we consider the range from μ - 3 to μ + 3. In this case, it means the range from 50 - 3 to 50 + 3.

For X = 47:
z-score = (47 - 50) / 1.67 = -1.80 (approximately)

For X = 53:
z-score = (53 - 50) / 1.67 = 1.80 (approximately)

Using the standard normal distribution table or software, we can find the area under the curve between -1.80 and 1.80, which represents the probability. It is approximately 0.935 (or 93.5%).

a. Sample mean not given, expect normality.

b. Without more info, expect sample mean = pop. mean

c. SEm = SD/√(n-1), but you can just use n

d, e, f. Z = (score-mean)/SEm

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