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 to be 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 1 / 3

Part 2 of 3 (b) Estimate the percentage of customer accounts for which the number of days is between 16 and 44. (Choose one) of the customer accounts have payment made between 16 and 44 days.

Solution

To solve the problem, we use the properties of a normal distribution: The average number of days \( \mu \) is 30, and the standard deviation \( \sigma \) is 7. For part (b), we need to find the percentage of customer accounts where the number of days is between 16 and 44. First, calculate the z-scores for 16 and 44. The z-score formula is: \[ z = \frac{x - \mu}{\sigma} \] Calculate the z-score for 16: \[ z_{16} = \frac{16 - 30}{7} = \frac{-14}{7} = -2 \] Calculate the z-score for 44: \[ z_{44} = \frac{44 - 30}{7} = \frac{14}{7} = 2 \] Next, check a standard normal distribution table (z-table) to find the probabilities corresponding to these z-scores. - The probability for \( z = -2 \) is approximately 0.0228. - The probability for \( z = 2 \) is approximately 0.9772. To find the percentage of customer accounts between 16 and 44 days, subtract the probabilities: \[ 0.9772 - 0.0228 = 0.9544 \] Convert to a percentage: \[ 0.9544 \times 100 = 95.44\% \] Approximately 95.44% of the customer accounts have payment made between 16 and 44 days.