Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.


SAT Italian Subject Test The scores on the SAT Italian Subject Test for the 2018–2020 graduating classes are normally distributed, with a mean of 628 and a standard deviation of 110. Random samples of size 25 are drawn from this population, and the mean of each sample is determined.

Verified step by step guidance

1. Step 1: Recall the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large, regardless of the population's distribution. In this case, the population is already normally distributed, so the sampling distribution of the sample mean will also be normal.
2. Step 2: Identify the population mean (μ) and population standard deviation (σ) from the problem. Here, the population mean is μ = 628, and the population standard deviation is σ = 110.
3. Step 3: Calculate the mean of the sampling distribution of the sample mean. According to the CLT, the mean of the sampling distribution is equal to the population mean. Therefore, the mean of the sampling distribution is μₓ̄ = μ = 628.
4. Step 4: Calculate the standard deviation of the sampling distribution of the sample mean, also known as the standard error (SE). The formula for the standard error is: $\sigma$_ₓ̄=σn, where n is the sample size. Substitute σ = 110 and n = 25 into the formula to find the standard error.
5. Step 5: Sketch the graph of the sampling distribution. Since the sampling distribution is normal, draw a bell-shaped curve centered at the mean μₓ̄ = 628. Label the x-axis with values representing the mean and standard deviations (e.g., μₓ̄ ± σₓ̄, μₓ̄ ± 2σₓ̄, etc.).

Key Concepts

Central Limit Theorem (CLT)

The Central Limit Theorem states that the distribution of the sample means will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large (typically n ≥ 30). This theorem is fundamental in statistics as it allows for the use of normal probability techniques to make inferences about population parameters based on sample statistics.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means from a population. It is characterized by its mean, which equals the population mean, and its standard deviation, known as the standard error, which is the population standard deviation divided by the square root of the sample size. Understanding this concept is crucial for estimating population parameters and conducting hypothesis tests.

Mean and Standard Deviation of Sampling Distribution

For a given population with mean (μ) and standard deviation (σ), the mean of the sampling distribution of the sample mean is equal to μ, while the standard deviation of the sampling distribution (standard error) is calculated as σ/√n, where n is the sample size. In the context of the SAT Italian Subject Test, this means that for samples of size 25, the mean will remain 628, and the standard deviation will be 110/√25 = 22.