Graphical Analysis In Exercises 9 and 10, the graph of a population distribution is shown with its mean and standard deviation. Random samples of size 100 are drawn from the population. Determine which of the figures labeled (a)–(c) would most closely resemble the sampling distribution of sample means. Explain your reasoning.

The waiting time (in seconds) to turn left at an intersection

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Step 1: Understand the problem. The question asks us to determine which figure (a, b, or c) most closely resembles the sampling distribution of sample means for random samples of size 100 drawn from the population. The population distribution is shown in the first graph, with a mean (μ) of 16.5 seconds and a standard deviation (σ) of 11.9 seconds.

Step 2: Recall the Central Limit Theorem. According to the theorem, the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large (n ≥ 30). Since the sample size is 100, the sampling distribution of the sample mean will be normal regardless of the shape of the population distribution.

Step 3: Calculate the mean and standard deviation of the sampling distribution. The mean of the sampling distribution (μₓ̄) is equal to the population mean (μ), which is 16.5 seconds. 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). Use the formula:

σ_{x} = \frac{σ_{}}{\sqrt{n}}

Step 4: Compare the graphs. The first graph represents the population distribution, which is skewed to the right. The second graph represents a normal distribution with the same mean (16.5 seconds) and a smaller standard deviation, which matches the characteristics of the sampling distribution of sample means.

Step 5: Conclude that the second graph (b) most closely resembles the sampling distribution of sample means. This is because the sampling distribution is normal due to the Central Limit Theorem, and its mean and standard deviation align with the calculated values for the sampling distribution.

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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 is the probability distribution of all possible sample means from a population. When random samples of size 100 are drawn, the Central Limit Theorem states that this distribution will tend to be normally distributed, regardless of the population's distribution, especially as the sample size increases. This concept is crucial for understanding how sample means behave and how they can be used to make inferences about the population mean.

Central Limit Theorem (CLT)

The Central Limit Theorem is a fundamental statistical principle that states that the distribution of the sample means will approach a normal distribution as the sample size becomes large, typically n ≥ 30. This theorem allows statisticians to make inferences about population parameters using sample statistics, as it provides a basis for the normal approximation of the sampling distribution, even if the original population distribution is not normal.

Mean and Standard Deviation

The mean (μ) is the average of a set of values, representing the central point of a distribution, while the standard deviation (σ) measures the dispersion or spread of the values around the mean. In the context of the question, the mean waiting time is 16.5 seconds, and the standard deviation is 11.9 seconds. These parameters are essential for understanding the characteristics of the population distribution and how they influence the shape of the sampling distribution of sample means.

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