Continuous Random Variables
First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using-score transformations. Finally, it defines tail probabilities and illustrates how to find them.
Areas of Tails of Distributions
1. Find the value ofa. that yields the probability shown.
3. Find the value ofthat yields the probability shown.
5. Find the indicated value of. (It is easier to find and negate it.)
7. Find the value ofthat yields the probability shown, where is a normally distributed random variable with mean and standard deviation .
11. Scores on a national exam are normally distributed with meana. Find the score that is the and standard deviation . th percentile.
b. Find the score that is the th percentile.
13. The monthly amount of water used per household in a small community is normally distributed with meangallons and standard deviation gallons. Find the three quartiles for the amount of water used.
15. Scores on the common final
exam given in a large enrollment multiple section course were normally
distributed with mean
17. Tests of a new tire developed
by a tire manufacturer led to an estimated mean tread life of
19. The weights
21. All students in a large
enrollment multiple section course take common in-class exams and a
common final, and submit common homework assignments. Course grades are
assigned based on students' final overall scores, which are
approximately normally distributed. The department assigns a
23. A machine for fillinga. If the machine is set to deliver -liter bottles of soft drink delivers an amount to each bottle that varies from bottle to bottle according to a normal distribution with standard deviation liter and mean whatever amount the machine is set to deliver. liters (so the mean amount delivered is liters) what proportion of the bottles will contain at least liters of soft drink?
b. Find the minimum setting of the mean amount delivered by the machine so that at least of all bottles will contain at least liters.