To determine whether a metal lathe that produces machine bearings is properly adjusted, a random sample of 36 bearings is collected and the diameter of each is measured. If the standard deviation of the diameters of the bearings measured over a long period of time is $0.001 \mathrm{inch}$, what is the approximate probability that the mean diameter of the sample of 36 bearings will fall between ( $\mu$ $0.0001)$ and $(\mu+0.0001)$ inch where $\mu$ is the population mean diameter of the bearings?

Step 1 ：This problem involves the concept of sampling distribution of the mean. The Central Limit Theorem (CLT) states that if you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Step 2 ：In this case, we are given that the standard deviation of the population is 0.001 inch, and we are taking a sample of 36 bearings. We are asked to find the probability that the sample mean will fall between \(\mu - 0.0001\) and \(\mu + 0.0001\).

Step 3 ：Since the sample size is large, we can use the Central Limit Theorem to solve this problem. The standard deviation of the sample mean (also known as the standard error) can be found by dividing the standard deviation of the population by the square root of the sample size.

Step 4 ：Once we have the standard error, we can find the z-scores corresponding to \(\mu - 0.0001\) and \(\mu + 0.0001\). The z-score is a measure of how many standard deviations an element is from the mean. We can use the z-score to find the probability that the sample mean will fall within the given range.

Step 5 ：Using the given values, we find that the standard error is approximately 0.00016666666666666666.

Step 6 ：The z-scores corresponding to \(\mu - 0.0001\) and \(\mu + 0.0001\) are approximately -0.6000000000000001 and 0.6000000000000001 respectively.

Step 7 ：Using these z-scores, we find that the probabilities corresponding to these z-scores are approximately 0.27425311775007355 and 0.7257468822499265 respectively.

Step 8 ：The probability that the sample mean will fall within the given range is the difference of these two probabilities, which is approximately 0.4514937644998529.

Step 9 ：Final Answer: The approximate probability that the mean diameter of the sample of 36 bearings will fall between \(\mu - 0.0001\) and \(\mu + 0.0001\) inch is \(\boxed{0.4515}\).