Historical production data indicate that the diameter of a ball bearing is normally distributed with a mean of 0.525 cm and a standard deviation of 0.008 cm.Suppose that a sample of 16 ball bearings are randomly selected from a very large lot.Determine the probability that the average diameter of a sampled ball bearing is greater than 0.530 cm.
A)0.2324
B)0.4938
C)0.5062
D)0.0062

Question

Quizplus · Accepted Answer

We are given that:
    sample size, n = 16
    mean, μ = 0.525 cm
    standard deviation, σ = 0.008 cm
    
The Central Limit Theorem (CLT) states that the sample means of sufficiently large sample sizes from any population, regardless of the shape of the population distribution, will follow 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.

Mathematically, the distribution of sample means follows:

X ~ N(μ, σ/√n)
    
where X is the random variable for the sample mean.

Using z-score to find the probability that the average diameter of a sampled ball bearing is greater than 0.530 cm, we can calculate:

z = (X - μ) / (σ / √n) = (0.530 - 0.525) / (0.008 / √16) = 2.5
    
From standard normal distribution tables, we find P(Z > 2.5) ≈ 0.0062.

Therefore, the probability that the average diameter of a sampled ball bearing is greater than 0.530 cm is 0.0062 or approximately 0.62%. The best choice is D.

Historical Production Data Indicate That the Diameter of a Ball