Final answer:
To find the probability of a ball bearing having a diameter greater than 127 millimeters, we calculate the z-score and use it with the standard normal distribution. The z-score is found to be -1.5, and the corresponding probability is approximately 0.9332 after subtracting from 1 the probability of being less than 127 millimeters.
Step-by-step explanation:
The question refers to the normal distribution of the diameters of ball bearings. We are given the mean diameter (133 millimeters) and the variance (16), which implies a standard deviation of 4 millimeters (since variance is the square of the standard deviation). To find the probability that the diameter of a randomly selected bearing is greater than 127 millimeters, we convert the diameter of 127 millimeters to a z-score.
A z-score is calculated using the formula:
Z = (X - μ) / σ
where X is the value of interest (127 millimeters), μ (mu) is the mean (133 millimeters), and σ (sigma) is the standard deviation (4 millimeters). Thus:
Z = (127 - 133) / 4 = -1.5
We then use a standard normal distribution table or a calculator with statistical functions to find the probability corresponding to a z-score of -1.5. The table will give us the probability that a value is less than 127 millimeters; since we want the probability that a value is greater than 127 millimeters, we subtract this value from 1. The probability of a z-score greater than -1.5 is about 0.9332. Therefore, the probability that the diameter of a selected bearing is greater than 127 millimeters is approximately 0.9332.