Question: Minneapolis has an average police response time of 511 seconds (population ) from the time a 911 call is placed to the time an officer arrives on the scene. The standard deviation of a police response time is 37.82 seconds. Assume that the police response time is normally distributed. The city wants to know whether a particular precinct that uses a computerized dispatching system has response times that are faster than the average for the city as a whole. If it turns out they are faster, the city will update the rest of the precincts to the same dispatching system. The following is a random sample of response times (in seconds) from that precinct:
481 492 467 478 503 472 482 533 512 480
The null and alternative hypotheses are given by: Ho : population > or equal to 511 and H1: population is < 511. Use alpha = 0.05
a) Calculate the test statistic for a z-test. Then, sketch the sampling distribution of the mean under the null hypothesis, and locate where this sample's mean falls with respect to it.
b) Find the critical value. Based on this information, what do you conclude about the precinct that uses the computerized dispatching system?
c) Find the p-value. Based on this information, what do you conclude about the precinct that uses the computerized dispatching system?
d) Did you come to the same conclusion when you used the critical value ( in part b) and the p-value ( in part b) and the p-value ( in part c)? Should the conclusions be the same? Why or why not?  Edit

Answer: a.
x bar= (481+ 492 +467 +478 +503 +472 +482 +533 +512 +480)/10=490
Test statistic=(x bar-mu)/sd=(490-511)/(37.82/sqrt(10))=-1.75

b. at alpha = 0.05 critical z-value=+/-1.65
hence computed z (-1.75)< critical z-value (-1.65)
null hypothesis is rejected .
a computerized dispatching system has response times are not faster than the average for the city as a whole.
c. P(z<-1.75)=0.4599 a computerized dispatching system has response times are not faster than the average for the city as a whole.

d) yes.  Edit