The distribution of the number of siblings for students at a large high school is skewed to the right with mean 1.8 siblings and standard deviation 0.7 sibling. A random sample of 100 students from the high school will be selected, and the mean number of siblings in the sample will be calculated.Which of the following describes the sampling distribution of the sample mean for samples of size 100 ?
A. Skewed to the right with standard deviation 0.7 sibling
B. Skewed to the right with standard deviation less than 0.7 sibling
C. Skewed to the right with standard deviation greater than 0.7 sibling
D. Approximately normal with standard deviation 0.7 sibling
E. Approximately normal with standard deviation less than 0.7 sibling

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]

In this problem, we have that:

Skewed right distribution, with [tex]\mu = 1.8, \sigma = 0.7[/tex]

Sampling distribution of the sample mean for samples of size 100

By the Central Limit Theorem, they will be approximately normal, with mean [tex]\mu = 1.8[/tex], and standard deviation [tex]s = \frac{0.7}{\sqrt{100}} = 0.07[/tex]

So the correct answer is:

E. Approximately normal with standard deviation less than 0.7 sibling