Question

A sample survey of 51 discount brokers showed that the mean price charged for a trade of 100 shares at 50 per share was $33.78. The survey is conducted annually. With the historical data available, assume a known population standard deviation of $13. a. Using the sample data, what is the margin of error associated with a 90% confidence interval (to decimals)?

Answer 1

Using a z-score of 1.645 for a 90% confidence interval and the formula for margin of error with a known population standard deviation, the calculation results in a margin of error of approximately $2.99 for the survey of discount brokers.

Step-by-step explanation:

To calculate the margin of error for a 90% confidence interval, we use the formula for the margin of error when the population standard deviation is known:

E = z*(σ/√n)

where:
E is the margin of error,
z is the z-score that corresponds to the desired confidence level,
σ is the population standard deviation, and
n is the sample size.

First, we look up the z-score for a 90% confidence level, which is about 1.645. Then we plug in the known standard deviation of $13 and the sample size of 51:

E = 1.645 * ($13/√51)

Using a calculator, we compute the margin of error:

E = 1.645 * ($13/√51) ≈ 1.645 * ($13/7.141) ≈ 1.645 * $1.818 ≈ $2.99

Therefore, the margin of error associated with a 90% confidence interval for this sample survey is $2.99.