Question

Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene. Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 11.6 minutes and a standard deviation of 2.3 minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.)

(a) the response time is between 7 and 13 minutes


(b) the response time is less than 7 minutes


(c) the response time is more than 13 minutes

Answer 1

Solution :

Given that ,

mean = = 11.6

standard deviation = = 2.3

(a)

P(7 < x < 13) = P((7 - 11.6 / 2.3) < (x - ) / < (13 - 11.6) / 2.3) )

P(7 < x < 13) = P(-2< z < 0.61)

P(7 < x < 13) = P(z < 0.61) - P(z < -2)

P(7 < x < 13) = 0.7291 - 0.0228 = 0.7063

Probability = 0.7063

(b)

P(x < 7) = P((x - ) / < (7 - 11.6) / 2.3) = P(z < -2)

Using standard normal table,

P(x < 7) = 0.0228

Probability = 0.0228

(c)

P(x > 13) = 1 - P(x < 13)

= 1 - P((x - ) / < (13 - 11.6) / 2.3)

= 1 - P(z < 0.61)

= 1 - 0.7291

= 0.2709

P(x > 13) = 0.2709

Probability = 0.2709