Question

question 4
suppose students ages follow a skewed right distribution with a mean of 23 years old and a standard deviation of 4 years. if we randomly sample 200 students, which of the following statements about the sampling distribution of the sample mean age is incorrect?
the standard deviation of the sampling distribution is equal to 4 years.
none of the answers.
the shape of the sampling distribution is approximately normal.
the mean of the sampling distribution is approximately 23 years old.

Explanation:

Step1: Recall central - limit theorem

The central - limit theorem states that for a sample of size $n$ from any population (regardless of the shape of the population distribution), the sampling distribution of the sample mean $\bar{X}$ has mean $\mu_{\bar{X}}=\mu$ and standard deviation $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$, and when $n$ is large (usually $n\geq30$), the sampling distribution of the sample mean is approximately normal. Here, $\mu = 23$ (population mean), $\sigma = 4$ (population standard deviation), and $n = 200$.

Step2: Calculate the standard deviation of the sampling distribution

The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$. Substituting $\sigma = 4$ and $n = 200$, we get $\sigma_{\bar{X}}=\frac{4}{\sqrt{200}}\approx\frac{4}{14.14}\approx0.283
eq4$.

Step3: Analyze the mean and shape of the sampling distribution

The mean of the sampling distribution of the sample mean is $\mu_{\bar{X}}=\mu = 23$. Since $n=200\geq30$, the shape of the sampling distribution of the sample mean is approximately normal.

Answer:

The standard deviation of the sampling distribution is equal to 4 years.