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.

Renewable Energy The zloty is the official currency of Poland. During a recent period of two years, the day-ahead prices for renewable energy in Poland (in zlotys per mega-watt hour) have a mean of 158.51 and a standard deviation of 33.424. Random samples of size 100 are drawn from this population, and the mean of each sample is determined. (Adapted from Multidisciplinary Digital Publishing Institute)

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Step 1: Recall the Central Limit Theorem (CLT). The CLT states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution. The mean of the sampling distribution will equal the population mean, and the standard deviation of the sampling distribution (known as the standard error) will be the population standard deviation divided by the square root of the sample size.

Step 2: Identify the given values from the problem. The population mean (μ) is 158.51, the population standard deviation (σ) is 33.424, and the sample size (n) is 100.

Step 3: Calculate the mean of the sampling distribution. According to the CLT, the mean of the sampling distribution of sample means is the same as the population mean. Therefore, the mean of the sampling distribution is μ = 158.51.

Step 4: Calculate the standard error of the sampling distribution. The formula for the standard error (SE) is:

SE = \frac{σ}{\sqrt{n}}

. Substitute the given values:

SE = \frac{33.424}{\sqrt{100}}

. Simplify the denominator to find the standard error.

Step 5: Sketch the graph of the sampling distribution. The graph will be a normal distribution centered at the mean (158.51) with a standard deviation equal to the calculated standard error. Label the x-axis with values around the mean, spaced by increments of the standard error, and indicate the bell-shaped curve.

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Key Concepts

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

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

A sampling distribution is the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. It describes how the sample means vary from sample to sample and is crucial for understanding the variability and reliability of estimates derived from sample data.

Mean and Standard Deviation of Sampling Distribution

The mean of the sampling distribution of sample means is equal to the population mean, while the standard deviation (known as the standard error) is calculated by dividing the population standard deviation by the square root of the sample size (σ/√n). These measures help quantify the expected value and variability of sample means, which are essential for making statistical inferences.

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