6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:23 minutesProblem 5.4.9
Textbook Question
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
Verified step by step guidance1
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: .
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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