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 = 1275, sigma =6, n = 1000

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 = 1275) into the formula. This means the mean of the sampling distribution is also 1275.
3. Step 3: Recall the formula for the standard deviation of the sampling distribution of sample means. The standard deviation of the sampling distribution (denoted as σₓ̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). The formula is: σₓ̄ = σ / √n.
4. Step 4: Substitute the given values into the formula for σₓ̄. Use σ = 6 and n = 1000. This gives: σₓ̄ = 6 / √1000.
5. Step 5: Simplify the expression for σₓ̄ by calculating the square root of 1000 and dividing 6 by that value. This will give the standard deviation of the sampling distribution.

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. The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution will approach a normal distribution as the sample size increases.

Mean of the Sampling Distribution

The mean of the sampling distribution of the 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 = 1275, the mean of the sampling distribution will also be 1275.

Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution, known as the standard error, measures the dispersion 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 = 6 and n = 1000, the standard error can be computed to understand how much variability to expect in the sample means.