A manager at a local manufacturing company has been monitoring the output of one of the machines used to manufacture chromium shells.Past data indicate that if the machine is functioning properly,the length of the shells produced by this machine can be modeled as being normally distributed with a mean of 118 centimeters and a standard deviation of 6.3 centimeters.Suppose 10 shells produced by this machine are randomly selected.What is the probability that the average length of these 10 shells will be between 116 and 120 centimeters when the machine is operating "properly"?
A)0.2709
B)0.2943
C)0.6826
D)0.9656

Question

Quizplus · Accepted Answer

To find the probability that the average length of these 10 shells will be between 116 and 120 centimeters, we use the Central Limit Theorem, which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough (n ≥ 30 is a common rule of thumb, but smaller sizes can often still be effectively modeled with normal distribution, especially when the population distribution is known to be normal, as in this case). Here, the sample size is 10, and we're given that the population distribution is normal with a mean (μ) of 118 cm and a standard deviation (σ) of 6.3 cm. The standard error of the mean (σ_x̄) for the sample is σ/√n = 6.3/√10. We're interested in the probability that the sample mean is between 116 and 120, which corresponds to finding the area under the normal distribution curve between these two points. This is essentially looking for P(116 < X̄ < 120). Given the symmetry and properties of the normal distribution, and knowing that ±1 standard deviation from the mean includes about 68% of the data (Empirical Rule), the probability that the sample mean falls within this range is approximately 68.26%, since 116 and 120 are symmetrically 2 standard deviations away from the mean, capturing the central portion of the distribution.

A Manager at a Local Manufacturing Company Has Been Monitoring