In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

The population densities in people per square mile in the 50 U.S. states have a mean of 199.6 and a standard deviation of 265.4. Random samples of size 35 are drawn from this population, and the mean of each sample is determined.

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Step 1: Recall the formulas for the mean and standard deviation of the sampling distribution of sample means. The mean of the sampling distribution (μₓ̄) is equal to the population mean (μ), and the standard deviation of the sampling distribution (σₓ̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n).

Step 2: Substitute the given values into the formulas. The population mean (μ) is 199.6, the population standard deviation (σ) is 265.4, and the sample size (n) is 35. Use the formula for the standard deviation of the sampling distribution: σₓ̄ = σ / √n.

Step 3: Simplify the expression for the standard deviation of the sampling distribution. Calculate the square root of the sample size (√n) and divide the population standard deviation (σ) by this value.

Step 4: The mean of the sampling distribution (μₓ̄) is the same as the population mean, so μₓ̄ = 199.6. The standard deviation of the sampling distribution (σₓ̄) is the value obtained in Step 3.

Step 5: To sketch the graph of the sampling distribution, draw a normal distribution curve centered at the mean (199.6) with the standard deviation (σₓ̄) calculated in Step 3. Label the x-axis with values around the mean, spaced by increments of the standard deviation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sampling Distribution of Sample Means

The sampling distribution of sample means refers to the distribution of the means of all possible samples of a specific size drawn from a population. It is crucial for understanding how sample means behave, particularly in relation to the population mean. According to the Central Limit Theorem, as the sample size increases, the sampling distribution approaches a normal distribution, regardless of the population's distribution, provided the sample size is sufficiently large.

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. In this case, it is 199.6 people per square mile. This concept is essential for predicting the average outcome of sample means and is foundational for inferential statistics, allowing statisticians to make generalizations about the population based on sample data.

Standard Deviation of the Sampling Distribution (Standard Error)

The standard deviation of the sampling distribution, often referred to as the standard error, measures the variability of sample means around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. In this scenario, the standard error helps assess how much the sample means are expected to fluctuate, providing insight into the reliability of the sample estimates.

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