Question

Determining Sample Size. In Exercises 19–22, assume that each sample is a simple random sample obtained from a normally distributed population. Use Table 7-2 on page 338 to find the indicated sample size.

Statistics Student Incomes You want to estimate the standard deviation of the annual incomes of all current statistics students. Find the minimum sample size needed to be 95% confident that the sample standard deviation is within 20% of the population standard deviation. Are those incomes likely to satisfy the requirement of a normal distribution?

Short answer

The value of the minimum sample size to be 95% confident that the sample standard deviation is within 20% of the value of the population standard deviation $\left(\sigma \right)$is equal to 48.

With more values of incomes on the left side and few values towards the right side of the graph, the distribution of the annual incomes of the students will not be normally distributed but will be skewed to the right.

Step-by-step solution

Given information 

It is given that there should be 95% confidence that the sample standard deviation should be within 20% of the value of the population standard deviation. The sample size is needed to be determined.

Step 2:Determine the sample size 

Using the table given in the book, the minimum sample size required to be 95% confident that the sample standard deviation should be within 20% of the value of the population standard deviation is equal to 48.

Therefore, the required sample size is equal to 48.

Step 3:Normality of the population

The annual incomes of students tend to be quite low, with a few exceptions of high incomes.

Therefore, with more values on the left side and a tail on the right side, the distribution of the annual incomes of the students will be skewed to the right.

Thus, the distribution of the annual incomes of the students will not be normally distributed.