Law Enforcement: Police Response Time Police response time...

Law Enforcement: Police Response Time 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 (based on information from The Denver Post). Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 8.4 minutes and a standard deviation of 1.7 minutes. For a randomly received emergency call, what is the probability that the response time will be (a) between 5 and 10 minutes? (b) less than 5 minutes? (c) more than 10 minutes?

Law Enforcement: Police Response Time 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 (based on information from The Denver Post). Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 8.4 minutes and a standard deviation of 1.7 minutes. For a randomly received emergency call, what is the probability that the response time will be
(a) between 5 and 10 minutes?
(b) less than 5 minutes?
(c) more than 10 minutes?

Key Concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for the normal distribution calculates the probability that a random variable will have a value less than or equal to a given value. This function is essential for finding probabilities in a normal distribution and is used in combination with the Z-score transformation to determine probabilities for ranges of values.

Probability Calculations for Ranges and Tails

Calculating probabilities for a range involves finding the difference between the cumulative probabilities at two endpoints, and calculating tail probabilities involves either finding the cumulative probability directly or applying the complement rule (for instance, one minus the cumulative probability). These methods are fundamental when assessing the likelihood of events falling between or beyond certain thresholds in any normal distribution.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, forming a bell-shaped curve. It is characterized by its mean (average) and standard deviation (spread), and is widely used in statistics to represent real-valued random variables whose distributions are not known. Its properties allow for the determination of probabilities concerning outcomes by means of the cumulative distribution function.

Z-Score Transformation

The Z-score transformation converts a value from a normal distribution into a standardized score that indicates how many standard deviations that value is from the mean. This transformation allows one to use the standard normal distribution, with a mean of zero and a standard deviation of one, to compute probabilities associated with specific values by referring to standard normal distribution tables or software.

Transcript

0:00 28.

00:02 Police response time.

00:04 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.

00:13 Over a long period of time, it has been determined that the police response has a normal distribution with the mean response time of 8 .4 minutes, with a standard deviation of 1 .7 minutes.

00:25 For a randomly received emergency call, what is the probability that the response time will be a between, 5 and 10 minutes, b, less than 5 minutes, and c, more than 10 minutes.

00:48 So in order to set this up, i want to draw the picture.

00:52 So on my normal curve, the mean that's given to us is the center at 8 .4.

00:57 And i want to space out the take marks by 1 .7, which is my standard deviation.

01:05 So one center deviation to the right is at 10 .1.

01:10 Do that again to get my second standard deviation, 11 .8.

01:13 And my third is 13 .5.

01:17 I want to get my standard deviations to the left.

01:21 Get 6 .7, 5 and 3 .3.

01:28 Once i have this, i can now tackle the problems.

01:33 And a, the probability that a randomly selected call will be between 5 and 10 minutes.

01:40 So i want to identify 5 and 10, which is slightly under that first standard deviation to the of the mean.

01:50 And i want to find the region between these two.