Step 1 :Calculate the z-score for a pitch speed of 92.8 mph using the formula \(Z = \frac{X - \mu}{\sigma}\). Substituting the given values, we get \(Z = \frac{92.8 - 92.12}{2.43} = 0.28\).
Step 2 :To find the fraction of pitches between 90.4 mph and 92.8 mph, first calculate the z-scores for these speeds. \(Z_{90.4} = \frac{90.4 - 92.12}{2.43} = -0.71\) and \(Z_{92.8} = \frac{92.8 - 92.12}{2.43} = 0.28\). The corresponding probabilities from a standard normal distribution table are 0.239 and 0.610 respectively. The fraction of pitches between these two speeds is the difference of these two probabilities, \(0.610 - 0.239 = 0.371\).
Step 3 :To find the fraction of pitches above 92.8 mph, subtract the probability corresponding to the z-score of 92.8 mph from 1. This gives us \(1 - 0.610 = 0.390\).
Step 4 :To find the speed corresponding to the fastest 12% of pitches, first find the z-score corresponding to 0.12 from a standard normal distribution table, which is approximately 1.175. Then use this z-score to find the corresponding pitch speed using the formula \(X = \mu + Z\sigma\). Substituting the given values, we get \(X = 92.12 + 1.175*2.43 = 94.99\) mph. Rounded to the nearest 0.1 mph, the speed must be \(\boxed{95.0}\) mph.