An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What total sample size would the economist need to use for a 95% confidence interval if the width of the interval should not be more than $100?
A) n = 385
B) n = 1537
C) n = 40
D) n = 20

Question

Quizplus · Accepted Answer

To achieve a confidence interval width of $100 with a known population standard deviation of $1,000, the sample size can be calculated using the formula for the margin of error: $ E = Z \times \frac{\sigma}{\sqrt{n}} $. For a 95% confidence level, $ Z \approx 1.96 $. Solving for $ n $ when $ E = 50 $ (half of the total width of $100), we get $ n = \left(\frac{1.96 \times 1000}{50}\right)^2 = 1536.64 $, which rounds up to 1537.

An Economist Is Interested in Studying the Incomes of Consumers