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 100 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. graph most closely resembles the sampling distribution of the sample means, because $mu_{\bar{x}}=square$, $sigma_{\bar{x}}=square$, and the graph (type an integer or a decimal.)

Explanation:

Step1: Recall the Central - Limit Theorem

According to the Central - Limit Theorem, for a sample of size $n$ from any population with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means $\bar{X}$ has mean $\mu_{\bar{X}}=\mu$ and standard deviation $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$.

Step2: Identify the population parameters

Let's assume the population mean $\mu = 19.5$ and population standard deviation $\sigma = 11.5$, and the sample size $n = 100$.

Step3: Calculate the mean of the sampling distribution of the sample means

The mean of the sampling distribution of the sample means is $\mu_{\bar{X}}=\mu=19.5$.

Step4: Calculate the standard deviation of the sampling distribution of the sample means

We know that $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$. Substituting $\sigma = 11.5$ and $n = 100$ (so $\sqrt{n}=10$), we get $\sigma_{\bar{X}}=\frac{11.5}{10}=1.15$.

Step5: Match the graph

We look for a graph with mean $\mu_{\bar{X}} = 19.5$ and standard deviation $\sigma_{\bar{X}}=1.15$. Graph (b) has $\mu = 19.5$ and $\sigma = 1.15$.

Answer:

Graph (b) most closely resembles the sampling distribution of the sample means, because $\mu_{\bar{X}} = 19.5$, $\sigma_{\bar{X}}=1.15$, and the graph has the correct mean and standard - deviation values for the sampling distribution of the sample means.