A manufacturer knows that their items have a lengths that are approximately normally distributed, with a mean of 17.4 inches, and standard deviation of 0.5 inches. If 34 items are chosen at random, what is the probability that their mean length is greater than 17.5 inches? (Round answer to four decimal places)

Step 1 ：We are given that the lengths of the items are normally distributed with a mean of 17.4 inches and a standard deviation of 0.5 inches. We are asked to find the probability that the mean length of 34 randomly chosen items is greater than 17.5 inches.

Step 2 ：We can use the Central Limit Theorem which states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the shape of the population distribution.

Step 3 ：The Central Limit Theorem also tells us that the mean of the sample means will be the population mean, and the standard deviation of the sample means (also known as the standard error) will be the population standard deviation divided by the square root of the sample size.

Step 4 ：We calculate the standard error as \(\frac{0.5}{\sqrt{34}}\), which is approximately 0.08574929257125441.

Step 5 ：We calculate the z-score, which is the number of standard deviations a data point is from the mean. The z-score for a value x is calculated by \(\frac{x - \text{mean}}{\text{standard deviation}}\). Substituting the given values, we get a z-score of approximately 1.1661903789690768.

Step 6 ：We use a z-table or a function like scipy's norm.cdf to find the probability that a randomly chosen data point will be less than x. Since we want the probability that the mean length is greater than 17.5, we need to subtract this probability from 1. The probability is approximately 0.1218.

Step 7 ：Final Answer: The probability that the mean length of 34 randomly chosen items is greater than 17.5 inches is \(\boxed{0.1218}\).