An industrial fan is designed to move 2500 ft3of air per minute, on average.Under normal operating conditions, the standard deviation of air volume moved by this fan is 300 ft3/min.If the fan is operating as designed, what is the probability that the average of 30 randomly selected measurements of fan capacity will be less than 2000 ft3/min?
A) 0.0475
B) 0.0000
C) 0.9525
D) 0.913

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate the distribution of sample means. The mean of the sample means will be equal to the population mean (2500 ft3/min), and the standard deviation of the sample means will be equal to the standard deviation of the population (300 ft3/min) divided by the square root of the sample size (sqrt(30)).

z = (x̄ - μ) / (σ / sqrt(n))
z = (2000 - 2500) / (300 / sqrt(30))
z = -10.392

Using a standard normal distribution table, we find that the probability of z being less than -10.392 is essentially 0. Therefore, the probability that the average of 30 randomly selected measurements of fan capacity will be less than 2000 ft3/min is 0.

An Industrial Fan Is Designed to Move 2500 Ft3of Air