Answer: To solve this problem, we can use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means from a large enough sample follows a normal distribution, regardless of the shape of the population distribution, as long as the sample size is sufficiently large.
Given:
Mean diameter of steel bolts = 130 millimeters
Standard deviation = 6 millimeters
Sample size = 49
Sample mean we are interested in = 132.3 millimeters
To calculate the probability that the sample mean would be greater than 132.3 millimeters, we need to calculate the z-score and then find the corresponding probability from the standard normal distribution table.
First, we calculate the standard error (SE) using the formula:
SE = Standard deviation / √(Sample size)
SE = 6 / √49
SE = 6 / 7
SE ≈ 0.8571
Next, we calculate the z-score using the formula:
z = (Sample mean - Population mean) / SE
z = (132.3 - 130) / 0.8571
z ≈ 3.0995
Using the z-score table or a calculator, we find that the probability corresponding to a z-score of 3.0995 is approximately 0.9987.
Therefore, the probability that the sample mean would be greater than 132.3 millimeters is approximately 0.9987 (or 99.87% when rounded to four decimal places).