Chapter 6: Q4 (page 272)

Annual Incomes Annual incomes are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a large (n>30) random sample of annual incomes. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is large? Why or why not?

Short Answer

Expert verified

No, the distribution of incomes in the sample cannot be approximated by a normal distribution because the sample is large as the sample observations follow the distribution of the population.

Step by step solution

Given information

The population of annual incomes is skewed to the right. A large sample of size greater than 30 is selected from this population.

Distribution of a sample vs sampling distribution of sample mean

The distribution of sample observations will always be the same as that of the population.

The sampling distribution of sample mean can be assumed to be normally distributed if the population is normally distributed or the sample size is large.

Here, the population of annual incomes has a distribution that is skewed to the right and not normally distributed. A sample of size greater than 30 is selected from this population.

Thus, the distribution of incomes in this sample will also be skewed to the right and not normally distributed.

Further, the distribution of the sample means of such samples (size greater than 30) can be considered to be normally distributed as they are sufficiently large size.