Step 1 :First, we need to find the z-scores for each value. The z-score is calculated as \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 2 :For \(x = 82.88\), we have \(z = \frac{82.88 - 74.27}{8.61} = \frac{8.61}{8.61} = 1\).
Step 3 :Now, we need to find the area to the right of \(z = 1\) in the standard normal distribution table. The table value for \(z = 1\) is 0.3413, which represents the area to the left of \(z = 1\). To find the area to the right, we subtract this value from 1: \(1 - 0.3413 = 0.6587\).
Step 4 :To find the percentage of customers that spent more than \$82.88 per purchase, we multiply the area by 100: \(0.6587 \times 100 = 65.87\%\).
Step 5 :\(\boxed{65.9\%}\) of customers spent more than \$82.88 per purchase.
Step 6 :For \(x = 100.10\), we have \(z = \frac{100.10 - 74.27}{8.61} = \frac{25.83}{8.61} \approx 3\).
Step 7 :Now, we need to find the area to the left of \(z = 3\) in the standard normal distribution table. The table value for \(z = 3\) is 0.9987.
Step 8 :To find the percentage of customers that spent less than \$100.10 per purchase, we multiply the area by 100: \(0.9987 \times 100 = 99.87\%\).
Step 9 :\(\boxed{99.9\%}\) of customers spent less than \$100.10 per purchase.