Respond to the following in a minimum of 175 words:
Consider the demonstration problem 6.3 which uses a normal distribution to determine the probability associated with generating between 3.6 and 5 pounds of waste per year.
Discuss any one of the following concepts associated with this problem.
- What are the “clues” that the problem can be solved with the use of the normal distribution?
- What is the relationship between the x value (e.g 3.6) and the z score?
- How is the probability of the event (3.6 to 5 pounds of waste) related to a specific area under the curve?
- 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 < x < 5.00 | μ = 4.43 and σ = 1.32 = ?
Figure 6.13 displays 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 two x values in this problem (x1 = 3.60 and x2 = 5.00). The z formula can handle only one x value 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 each z value:
z = x − μ σ = 3.60 − 4.43 1.32 = − 0.63 z = x − μ σ = 5.00 − 4.43 1.32 = 0.43
FIGURE 6.13 Graphical Depiction of the Waste-Generation Problem with 3.60 < x < 5.00
Next, we look up each z value in the z distribution table. Since the normal distribution is symmetrical, the probability associated with z = −0.63 is the same as the probability associated with z = 0.63. Looking up z = 0.63 in the table yields a probability of .2357. The probability associated with z = 0.43 is .1664. Using these two probability values, we can get the probability that 3.60 < x < 5.00 by summing the two areas:
P ( 3.60 < x < 5.00 | μ = 4.43 and σ = 1.32 ) = .2357 + .1664 = .4021
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 6.14 displays the solution to this problem.
FIGURE 6.14 Solution of the Waste-Generation Problem with 3.60 < x < 5.00