Question

Provide an appropriate response.

The distribution of incomes of employees at one company is strongly skewed to the right and the mean and standard deviation of the incomes are known. Is it possible to determine the probability that a randomly selected employee earns more than $80,000? Is it possible to determine the probability that the mean income for 10 randomly selected employees is more than $80,000? Is it possible to determine the probability that the mean income for 50 randomly selected employees is more than $80,000? Explain your responses.

Answer 1

It is not possible to determine the probability that a randomly selected employee earns more than $80,000 because the distribution is not normal here. It is also not possible to determine the probability that the mean income for 10 randomly selected employees is more than $80,000, because the sample size here is 10 and Central Limit Theorem (CLT) will not be valid as it requires a sample size of 30 to make the distribution of sample means to be normal. Although, CLT can work with smaller samples too, but only if the population data is symmetric, which is not the case here.

Although the data is strongly skewed to the right, we can still determine the probability that means income for 50 randomly selected employees is more than $80,000. This is possible because of the Central Limit Theorem which states that regardless of the shape of the population distribution, the distribution of sample means will be approximately normal (here, sample size being 50). The Central Limit Theorem requires sample size to be at least 30 or even less if the data is symmetric with fewer outliers