Question

The number of siblings an individual has varies from student to student. The distribution of the number of siblings is strongly skewed to the right. The central limit theorem says that:________

(a) as we look at more and more students, their mean number of siblings gets ever closer to the mean u for all students.

(b) the mean number of siblings for any number of students has a distribution of the same shape (strongly skewed) as the distribution for individual students.

(c) the mean number of siblings for any number of students has a distribution that is close to Normal.

(d) the mean number of siblings for a large number of students has a distribution of the same shape (strongly skewed) as the distribution for individual students.

(e) the mean number of siblings for a large number of students has a distribution that is close to Normal.

Answer 1

The correct answer to the student's question is that the mean number of siblings for a large number of students has a distribution that is close to normal, as per the Central Limit Theorem.

Step-by-step explanation:

The Central Limit Theorem (CLT) is key in understanding the behavior of sample means when drawing samples from a population, regardless of the population's original distribution shape. According to the CLT, as the sample size increases, the sampling distribution of the sample mean will approach a normal distribution. This holds true even if the population distribution is not normal, such as the strongly skewed distribution of the number of siblings that students have.

So, the correct answer to the student's question is (e) the mean number of siblings for a large number of students has a distribution that is close to Normal. This phenomenon occurs because the means of large samples are less variable and more normally distributed than the individual observations in the population.

Additionally, it's important to distinguish between the mean of sample means and the distribution shape. The mean of the sampling distribution is equal to the population mean, and the standard deviation of the sample means (standard error) decreases as the sample size increases.

Answer 2

(e) the mean number of siblings for a large number of students has a distribution that is close to Normal.

Step-by-step explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

By the Central Limit Theorem

The sampling distributions with a large number of students(at least 30) will be approximately normal, so the correct answer is given by option e.