An economist is interested in studying the incomes of consumers in a particular country. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in a mean 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 = 1537
B) n = 385
C) n = 40
D) n = 20

Question

Quizplus · Accepted Answer

We can use the formula for margin of error to solve for the necessary sample size:

Margin of Error = z * standard deviation / sqrt(sample size)

Since we want to have a margin of error no more than $100, we can set up the following inequality:

100 = z * 1000 / sqrt(n)

Solving for n:

n = (z * 1000 / 100)^2

We know that for a 95% confidence interval, z = 1.96. Substituting this value:

n = (1.96 * 1000 / 100)^2

n = 1537.6, which we round up to 1537.

Therefore, the economist would need a total sample size of 1537 individuals in order to construct a 95% confidence interval with a width no more than $100. Choice A) is the correct answer.

An Economist Is Interested in Studying the Incomes of Consumers