Final answer:
The probability that the sample mean of 16 randomly selected Ping-Pong balls is less than 1.28 inches is approximately 2.28%, calculated using the z-score of -2.00 obtained from the sampling distribution of the sample mean.
Step-by-step explanation:
The problem revolves around finding the probability that the sample mean of a normally distributed population is less than a certain value. Given the mean μ = 1.30 inches, a standard deviation σ = 0.04 inches, and a sample size n = 16, we first need to determine the sampling distribution of the sample mean. The sampling distribution of the sample mean will also be normally distributed with mean μ = 1.30 inches and a standard deviation σ/√n (standard error), where σ is the standard deviation of the population and n is the sample size.
To find the standard error, we calculate σ/√n = 0.04 inches/√16 = 0.01 inches. Next, we need to calculate the z-score for the sample mean 1.28 inches. The z-score is found using the formula z = (X - μ)/(standard error), so z = (1.28 - 1.30) / 0.01 = -2.00. Now, we can use the cumulative standard normal distribution to find the probability that the z-score is less than -2.00, which corresponds to the probability that the sample mean is less than 1.28 inches.
Following the standard normal distribution, the probability (p-value) associated with a z-score of -2.00 is approximately 0.0228, indicating a 2.28% chance that the sample mean will be less than 1.28 inches if a random sample of 16 Ping-Pong balls is selected. To find this probability, one would typically use standard normal distribution tables or statistical software.