Refer to the Journal of the Acoustical Society of America (Feb. 1986) study of auditory nerve response rates in cats, discussed in Exercise 4.107. A key question addressed by the research is whether rate changes (i.e., changes in number of spikes per burst of noise) produced by tones in the presence of background noise are large enough to detect reliably. That is, can the tone be detected reliably when background noise is present? In the theory of signal detection, the problem involves a comparison of two probability distributions. Let Y represent the auditory nerve response rate (i.e., the number of spikes observed) under two conditions: when the stimulus is background noise only (N) and when the stimulus is a tone plus background noise (T). The probability distributions for Y under the two conditions are represented by the density functions, f_N(y) and f_T(y), respectively, where we assume that the mean response rate under the background-noise only condition is less than the mean response rate under the tone-plus-noise condition, i.e., μ_N < μ_T. In this situation, an observer sets a threshold C and decides that a tone is present if Y < C and decides that no tone is present if. Assume that f_N(y) and f_T(y) are both normal density functions with means μ_N = 10.1. spikes per burst and μ_T = 13.6 spikes per burst, respectively, and equal variances.

a. For a threshold of C = 11 spikes per burst, find the probability of detecting the tone given that the tone is present. (This is known as the detection probability.)

b. For a threshold of C = 11 spikes per burst, find the probability of detecting the tone given that only background noise is present. (This is known as the probability of false alarm.)

c. Usually, it is desirable to maximize detection probability while minimizing false alarm probability. Can you find a value of C that will both increase the detection probability (part a) and decrease the probability of false alarm (part b)?

Data From Exercise 4.107:

Level of benzene at petrochemical plants. Benzene, a solvent  commonly used to synthesize plastics and found in  consumer products such as paint strippers and high-octane  unleaded gasoline, has been classified by scientists as a  leukemia-causing agent. Let Y be the level (in parts per  million) of benzene in the air at a petrochemical plant. Then Y can take on the values 0, 1, 2, 3,....1,000,000 and can be approximated by a Poisson probability distribution.  In 1978, the federal government lowered the maximum allowable level of benzene in the air at a workplace from 10 parts per million (ppm) to 1 ppm. Any industry in  violation of these government standards is subject to severe penalties, including implementation of expensive  measures to lower the benzene level.

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