Consider the demonstration problem 6.3 which uses a normal distribution to determine the probability associated withgenerating between 3.6 and 5 pounds of waste per year.

Discuss any one of the following concepts associated with this problem.

1. What are the "clues" that the problem can be solved with the use of the normal distribution?
2. What is the relationship between the x value (e.g3.6) and the z score?
3. How is the probability of the event(3.6 to 5 pounds of waste) related to a specific area under the curve?
4. How do we determine the area between 2 z values?

DEMONSTRATION PROBLEM 6.3

Using this same waste-generation example, if a U.S. person is randomly selected, what is the probability that the person generates between 3.60 and 5.00 pounds of waste per day?

We can summarize this problem as:

P(3.60

Figure 6.13displays a graphical representation of the problem. Note that the area under the curve for which we are solving crosses over the mean of the distribution. Note that there are twoxvalues in this problem (x₁= 3.60andx₂= 5.00). Thezformula can handle only onexvalue at a time. Thus, this problem needs to be worked out as two separate problems and the resulting probabilities added together. We begin the process by solving for eachzvalue:

z = x = 3.60 4.43 1.32 = 0.63 z

= x = 5.00 4.43 1.32 = 0.43

FIGURE 6.13Graphical Depiction of the Waste-Generation Problem with 3.60 <x< 5.00

Next, we look up eachzvalue in thezdistribution table. Since the normal distribution is symmetrical, the probability associated withz= 0.63is the same as the probability associated withz= 0.63. Looking upz= 0.63in the table yields a probability of .2357. The probability associated withz= 0.43is .1664. Using these two probability values, we can get the probability that3.60 <x< 5.00by summing the two areas:

P(3.60

The probability that a randomly selected person in the U.S. has between 3.60 and 5.00 pounds of waste generation per day is .4021 or 40.21%.Figure 6.14displays the solution to this problem.

FIGURE 6.14Solution of the Waste-Generation Problem with 3.60 <x< 5.00