A carload of steel rods has arrived at Cybermatic Construction Company. The car contains 50,000 rods. Claude Ong, manager of Quality Assurance, directs his crew measure the lengths of 100 randomly selected rods. If the population of rods have a mean length of 120 cm and a standard deviation of 0.05 cm, the probability that Claude's sample has a mean less than 119.985 cm is ___.
A) 0.9974
B) 0.0026
C) 0.4987
D) 0.0013
E) 0.0030

Question

Quizplus · Accepted Answer

We need to use the central limit theorem because we have a large population (50,000 rods) and a sample size (100 rods) that is less than 10% of the population. The central limit theorem states that the sample mean follows a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

So, the sample mean follows a normal distribution with a mean of 120 cm and a standard deviation of 0.005 cm (0.05 cm / sqrt(100)).

Using a z-score table, we can find that the z-score for a sample mean of 119.985 cm is:

z = (119.985 - 120) / 0.005 = -3

The probability of getting a sample mean less than 119.985 cm is the same as the probability of getting a z-score less than -3, which is approximately 0.0013 (from the z-score table).

Therefore, the answer is D) 0.0013.

A Carload of Steel Rods Has Arrived at Cybermatic Construction