91Ó°ÊÓ

A continuous random variable \(X\) has a normal distribution with mean 100 and standard deviation \(10 .\) Sketch a qualitatively accurate graph of its density function.

Scores on the common final exam given in a large enrollment multiple section course were normally distributed with mean 69.35 and standard deviation \(12.93 .\) The department has the rule that in order to receive an A in the course his score must be in the top \(10 \%\) of all exam scores. Find the minimum exam score that meets this requirement.

The average finishing time among all high school boys in a particular track event in a certain state is 5 minutes 17 seconds. Times are normally distributed with standard deviation 12 seconds. a. The qualifying time in this event for participation in the state meet is to be set so that only the fastest \(5 \%\) of all runners qualify. Find the qualifying time. (Hint: Convert seconds to minutes.) b. In the western region of the state the times of all boys running in this event are normally distributed with standard deviation 12 seconds, but with mean 5 minutes 22 seconds. Find the proportion of boys from this region who qualify to run in this event in the state meet.

A regulation hockey puck must weigh between 5.5 and 6 ounces. In an alternative manufacturing process the mean weight of pucks produced is 5.75 ounce. The weights of pucks have a normal distribution whose standard deviation can be decreased by increasingly stringent (and expensive) controls on the manufacturing process. Find the maximum allowable standard deviation so that at most 0.005 of all pucks will fail to meet the weight standard.