Final answer:
To determine the probability that the sample mean diameter of steel bolts is less than 137.3 millimeters, one must calculate the z-score and then refer to the standard normal distribution. With the provided mean, standard deviation, and sample size, the z-score can be found, and the corresponding probability can be obtained from the normal distribution tables or tools.
Step-by-step explanation:
The question entails calculating the probability that the sample mean of steel bolts will be less than a given value, assuming the sample is drawn from a normally distributed population with known mean and standard deviation. To find this probability, we utilize the Central Limit Theorem which indicates that the sampling distribution of the sample means will be normal since the sample size is sufficiently large (n>30).
First, we calculate the standard error of the mean (SEM) using the formula SEM = σ/√n, where σ is the population standard deviation and n is the sample size. With a standard deviation (σ) of 5 millimeters and a sample size (n) of 41, the SEM equals 5/√41.
Next, we calculate the z-score for 137.3 millimeters using the formula z = (X - μ)/SEM, where X is the sample mean, and μ is the population mean. Once we have the z-score, we can use standard normal distribution tables or calculators to find the probability associated with this z-score. The correct answer will thus reveal the probability that the sample mean diameter of steel bolts will be less than 137.3 millimeters