In Exercises 1–4, a population has a mean mu and a standard deviation sigma. Find the mean and standard deviation of the sampling distribution of sample means with sample size n.


Mu = 790, sigma =48, n = 250

Verified step by step guidance

1. Step 1: Recall the formula for the mean of the sampling distribution of sample means. The mean of the sampling distribution (denoted as μₓ̄) is equal to the population mean (μ). Therefore, μₓ̄ = μ.
2. Step 2: Substitute the given value of the population mean (μ = 790) into the formula. This means the mean of the sampling distribution is μₓ̄ = 790.
3. Step 3: Recall the formula for the standard deviation of the sampling distribution of sample means, also known as the standard error (SE). The formula is SE = σ / √n, where σ is the population standard deviation and n is the sample size.
4. Step 4: Substitute the given values into the formula for SE. Here, σ = 48 and n = 250. The formula becomes SE = 48 / √250.
5. Step 5: Simplify the expression for SE by calculating the square root of 250 and dividing 48 by that value. This will give you the standard deviation of the sampling distribution of sample means.

Key Concepts

Sampling Distribution

The sampling distribution of the sample mean is the probability distribution of all possible sample means from a population. It describes how the means of different samples will vary and is crucial for understanding the behavior of sample statistics. According to the Central Limit Theorem, as the sample size increases, the sampling distribution approaches a normal distribution, regardless of the population's distribution.

Mean of the Sampling Distribution

The mean of the sampling distribution of sample means, also known as the expected value, is equal to the population mean (mu). This means that if you take many samples and calculate their means, the average of those means will converge to the population mean. In this case, with mu = 790, the mean of the sampling distribution will also be 790.

Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution, known as the standard error, measures the variability of sample means around the population mean. It is calculated by dividing the population standard deviation (sigma) by the square root of the sample size (n). For this problem, with sigma = 48 and n = 250, the standard error can be computed to understand how much sample means will typically deviate from the population mean.