Final answer:
The 80th percentile for the time between call arrivals at a switchboard, where calls follow the exponential distribution at a mean of one every 15 seconds, is approximately 22.2 seconds.
Step-by-step explanation:
To find the 80th percentile for the time between call arrivals when calls arrive at a switchboard at a mean of one every 15 seconds and follow the exponential distribution, we can use the formula for the exponential distribution's percentile. The exponential distribution function is given by F(x) = 1 - e-λx where λ is the rate of occurrences, which in this case is 1/15, since a call arrives every 15 seconds. To find the 80th percentile (P80), we set the cumulative distribution function to 0.8: 0.8 = 1 - e-(1/15)x.
The next step is to solve for x: 0.2 = e-(1/15)x. Taking the natural logarithm of both sides, we get ln(0.2) = -(1/15)x. Solving for x gives x = -15 * ln(0.2). Calculating this gives an approximate value for the 80th percentile for the time between calls, which is equivalent to 22.2 seconds, which corresponds to option (a).