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 most 3.1 cm?
A)0.8413
B)1
C)0.7157
D)0.2843
E)0.1587

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. Here, the variance of the sample mean is 0.03 cm² / 3 = 0.01 cm², so the standard deviation of the sample mean is sqrt(0.01) = 0.1 cm. To find the probability that the sample mean is at most 3.1 cm, we calculate the Z-score for 3.1 cm: Z = (3.1 - 3) / 0.1 = 1. The probability of a Z-score being less than or equal to 1 in a standard normal distribution is 0.8413.

In a Manufacturing Process a Machine Produces Bolts