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 = 45, sigma =15, n = 100

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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.

Step 2: Substitute the given value of the population mean (μ = 45) into the formula. This means that the mean of the sampling distribution of sample means is also 45.

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.

Step 4: Substitute the given values into the formula for SE. Here, σ = 15 and n = 100. The formula becomes SE = 15 / √100.

Step 5: Simplify the expression for SE by calculating the square root of the sample size (√100 = 10) and dividing the population standard deviation by this value. This will give you the standard deviation of the sampling distribution of sample means.

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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 sample means vary from sample to sample 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, also known as the expected value of the sample mean, is equal to the population mean (mu). This means that if you take many samples from a population and calculate their means, the average of those sample means will converge to the population mean. In this case, with mu = 45, the mean of the sampling distribution is also 45.

Standard Deviation of the Sampling Distribution (Standard Error)

The standard deviation of the sampling distribution, often referred to 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 = 15 and n = 100, the standard error would be 15 / √100 = 1.5.

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 = 45, sigma =15, n = 100