Final answer:
The top 7% of customers spend at least $195.86, and the probability that a customer will spend at least $18 per month is 30.03%.
Step-by-step explanation:
To determine how much the top 7% of customers spend, we need to find the value of the 93rd percentile of the exponential distribution with a mean of $72. We can use the formula x = -ln(1-p) * m, where x is the value of the percentile, p is the desired percentile (0.93 in this case), and m is the mean. Substituting the values into the formula, we get x = -ln(1-0.93) * 72 = -ln(0.07) * 72 ≈ 195.86. So, the top 7% of customers spend at least $195.86.
To find the probability that a customer will spend at least $18 per month, we need to find the percentile corresponding to $18. Rearranging the formula, we get p = 1 - exp(-x/m), where p is the probability, x is the value of interest ($18 in this case), and m is the mean. Substituting the values into the formula, we get p = 1 - exp(-18/72) ≈ 0.3003. So, the probability that a customer will spend at least $18 per month is approximately 0.3003 or 30.03%.