In a manufacturing process a machine produces bolts. The lengths of the bolts follow a normal distribution with an average of 3 cm and a variance of 0.03 cm2. If we randomly select three bolts from this process:
-What is the probability the mean length of the bolt is at least 3.16 cm?
A)0.9772
B)0.0548
C)0.9452
D)0.4452
E)0.0228

Question

Quizplus · Accepted Answer

The distribution of the sample mean for a normally distributed variable is also normal with mean equal to the population mean (3 cm) and variance equal to the population variance divided by the sample size (0.03 cm² / 3). Thus, the standard deviation of the sample mean is sqrt(0.03/3) = sqrt(0.01) = 0.1 cm. To find the probability that the sample mean is at least 3.16 cm, we calculate the Z-score: (3.16 - 3) / 0.1 = 1.6. Using the Z-table, the probability of a Z-score less than 1.6 is approximately 0.9452, so the probability of being greater than 1.6 is 1 - 0.9452 = 0.0548.

In a Manufacturing Process a Machine Produces Bolts