Final answer:
The probability that 50 randomly selected pins will be within the stated tolerance levels is 0, indicating it is highly unlikely that all 50 pins will have a hardness of exactly 50.
Step-by-step explanation:
To find the probability that 50 randomly selected pins will be within the stated tolerance levels, we need to calculate the z-score of the value 50 with respect to the mean and standard deviation. The z-score formula is (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
In this case, the mean is 50 and the standard deviation is 1.6. Plugging in the values, we get (50 - 50) / 1.6 = 0. Therefore, the probability that 50 randomly selected pins will be within the stated tolerance levels is 0. This means that it is highly unlikely that all 50 pins will have a hardness of exactly 50.
The task involves applying statistics concepts such as mean, standard deviation, probability calculations, and the central limit theorem to describe the quality control of screw diameter sizes.
These calculations are crucial for evaluating manufacturing processes and ensuring that products meet specified tolerance levels.