Question: Pay your bills: In a large sample of customer accounts, a utility company determined that the…

Pay your bills: In a large sample of customer accounts, a utility company determined that the average number of days between when a bill was sent out and when the payment was made is 30 with a standard deviation of 7 days. Assume the data are approximately bell-shaped.

Part 1 of 3

(a) Between what two values will approximately 68% of the numbers of days be? Approximately 68% of the customer accounts have payment made between 23 and 37 days.

Part 2 of 3

(b) Estimate the percentage of customer accounts for which the number of days is between 16 and 44. Approximately 95% of the customer accounts have payment made between 16 and 44 days.

Part: 2 / 3

Part 3 of 3

(c) Estimate the percentage of customer accounts for which the number of days is between 9 and 51. (Choose one) of the customer accounts have payment made between 9 and 51 days.

Solution

The problem involves a normal distribution with a mean (\(\mu\)) of 30 days and a standard deviation (\(\sigma\)) of 7 days. For part (c), we need to estimate the percentage of customer accounts for which the number of days is between 9 and 51. First, calculate how many standard deviations 9 and 51 are from the mean. For 9 days: Find the z-score: \[ z = \frac{X - \mu}{\sigma} \] \[ z = \frac{9 - 30}{7} \] \[ z = \frac{-21}{7} \] \[ z = -3 \] For 51 days: Find the z-score: \[ z = \frac{51 - 30}{7} \] \[ z = \frac{21}{7} \] \[ z = 3 \] According to the empirical rule (68-95-99.7 rule) for a normal distribution: - About 68% of the data falls within 1 standard deviation of the mean. - About 95% falls within 2 standard deviations. - About 99.7% falls within 3 standard deviations. Since 9 and 51 days are each 3 standard deviations from the mean, approximately 99.7% of the customer accounts have payments made between these days.