Questions: The graph of the waiting time (in seconds) at a red light is shown below on the left with its mean and standard deviation. Assume that a sample size of 225 is drawn from the population. Decide which of the graphs labeled (a)-(c) would most closely resemble the sampling distribution of the sample means. Explain your reasoning. (a) (b) (c) Graph most closely resembles the sampling distribution of the sample means, because μx̄= , σx̄= , and the graph (Type an integer or a decimal.)

Solution Steps

The mean of the sample means (𝜇𝑥̅) is equal to the population mean (𝜇). From the original graph, we can see that the population mean is 17.2. Therefore, 𝜇𝑥̅ = 17.2.

The standard deviation of the sample means (𝜎𝑥̅) is equal to the population standard deviation (𝜎) divided by the square root of the sample size (n). From the original graph, we can see that 𝜎 = 12.6 and we are given that n = 225. Therefore, 𝜎𝑥̅ = 𝜎 / √n = 12.6 / √225 = 12.6 / 15 = 0.84.

We are looking for a graph with 𝜇𝑥̅ = 17.2 and 𝜎𝑥̅ = 0.84. Graph (a) has 𝜇𝑥̅ = 17.2 and 𝜎𝑥̅ = 12.6, graph (b) has 𝜇𝑥̅ = 1.1 and 𝜎𝑥̅ = 12.6 and graph (c) has 𝜇𝑥̅ = 1 and 𝜎𝑥̅ = 0. None of the given graphs exactly match our calculated values. However, the Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample means approaches a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size. Thus, graph (a) most closely resembles the distribution of sample means as it maintains the same mean and has a smaller standard deviation than the population. This reduction in the standard deviation is also why the graph appears taller. Since no graph shows the correct standard deviation, we will pick the closest graph.

Final Answer