Chattahoochee MATH1127: Introduction to Statistics (60022) Lesson 7.1 The Central Limit Theorem for Sample Means (Avera Unit 3 Chapter 7: Lesson 7.1 Assignment Score: $60 / 100 \quad 14 / 20$ answered Question 16 A manufacturer knows that their items have a normally distributed length, with a mean of 10.1 inches, and standard deviation of 0.5 inches. If 5 items are chosen at random, what is the probability that their mean length is less than 10.1 inches? Question Help: D Post to forum

Step 1 ：The problem is asking for the probability that the mean length of 5 randomly chosen items is less than 10.1 inches. This is a problem of 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 2 ：In this case, we know that the population is normally distributed with a mean (μ) of 10.1 inches and a standard deviation (σ) of 0.5 inches. We are taking a sample of 5 items (n=5).

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

Step 4 ：We are asked to find the probability that the sample mean is less than 10.1 inches. This is equivalent to finding the z-score for 10.1 inches and looking up this z-score in the standard normal distribution table.

Step 5 ：The z-score is calculated as \((X - μ) / (σ / \sqrt{n})\), where X is the value for which we want to find the z-score, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Step 6 ：In this case, X = 10.1, μ = 10.1, σ = 0.5, and n = 5. Let's calculate the z-score and find the corresponding probability.

Step 7 ：The calculated z-score is 0, which corresponds to a probability of 0.5 in the standard normal distribution. This means that there is a 50% chance that the mean length of 5 randomly chosen items is less than 10.1 inches. This makes sense because 10.1 inches is the mean of the population, so we would expect about half of the sample means to be less than the population mean and about half to be greater.

Step 8 ：Final Answer: The probability that the mean length of 5 randomly chosen items is less than 10.1 inches is \(\boxed{0.5}\) or \(\boxed{50\%}\).