The nozzle of a mixing vibrator is tested for its number of vibrations. The vibration frequency, for each nozzle sample, can be modeled by a normal distribution with mean 128 and standard deviation 16 PdM.

(a) If engineering specifications require the sample to have a vibration frequency of \(100 \mathrm{PdM}\), what is the probability that a sample will fail to meet specifications?

(b) In the long run, what proportion of samples will fail? Explain your answer.

(c) The mean vibration frequency can be increased by using different materials. What new mean is required, when the standard deviation is 16 , to reduce the probability of not meeting specifications to 0.05 ?