Question

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 64 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) Graph (1) most closely resembles the sampling distribution of the sample means, because μ xˉ​ = σ x = , and the graph (2) (Type an integer or a decimal.) (1) (b) (2) is the same shape as the graph for the original distribution (c) approximates a normal curve (a)

Answer 1

Graph (c) most closely resembles the sampling distribution of the sample means.

Step-by-step explanation:

The sampling distribution of the sample means is characterized by the central limit theorem, stating that for a sufficiently large sample size, the distribution of sample means will be approximately normal, regardless of the shape of the original population distribution.

In this case, the sample size is 64, which is considered large enough for the central limit theorem to apply.

Graph (c) approximating a normal curve aligns with the central limit theorem. When sample means are plotted, they tend to form a bell-shaped curve resembling a normal distribution.

This is because the means aggregate and smooth out the variations present in the original distribution. The fact that graph (c) closely resembles a normal curve suggests that it is more likely to represent the sampling distribution of the sample means.

On the other hand, graphs (a) and (b) do not exhibit the characteristics expected of a sampling distribution of sample means. Graph (a) appears to have a skewed distribution, and graph (b) lacks the symmetry and smoothness associated with a normal distribution.

Therefore, based on the principles of the central limit theorem, graph (c) is the most suitable representation of the sampling distribution of the sample means in this scenario.