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 27 with a standard deviation of 7 days. Assume the data to be approximately bell-shaped.

Question

(a) Between what two values will approximately 68% of the numbers of days be?

Approximately 68% of the customer accounts have payment made between

Answer 1

In a normal distribution (bell-shaped), approximately 68% of the data falls within one standard deviation of the mean.

Given:

• Mean (average) number of days = 27 days
• Standard deviation = 7 days

To find the range for approximately 68% of the days:

1. Lower value = Mean - Standard deviation = 27 - 7 = 20 days
2. Upper value = Mean + Standard deviation = 27 + 7 = 34 days

Thus, approximately 68% of the customer accounts have payments made between 20 days and 34 days.

Answer 2

(b) Estimate the percentage of customer accounts for which the number of days is between 13 and 41

Answer 3

To estimate the percentage of customer accounts for which the number of days between when a bill was sent out and when the payment was made is between 13 and 41 days, we can again use the properties of the normal distribution.

Given:

• Mean = 27 days
• Standard deviation = 7 days

First, we need to determine how many standard deviations away from the mean the values 13 and 41 are:

1. For the lower value (13 days): \[ z = \frac{(X - \mu)}{\sigma} = \frac{(13 - 27)}{7} = \frac{-14}{7} = -2 \]

2. For the upper value (41 days): \[ z = \frac{(X - \mu)}{\sigma} = \frac{(41 - 27)}{7} = \frac{14}{7} = 2 \]

Now, we look up these \( z \)-scores in the standard normal distribution.

• A \( z \)-score of -2 corresponds to approximately 2.5% in the lower tail.
• A \( z \)-score of 2 corresponds to approximately 97.5% in the upper tail.

To find the percentage of customer accounts within this range:

\[ \text{Percentage} = \text{Upper} - \text{Lower} = 97.5% - 2.5% = 95% \]

Thus, approximately 95% of customer accounts have payments made between 13 days and 41 days.

Answer 4

(c) Estimate the percentage of customer accounts for which the number of days is between 6 and 48

Answer 5

To estimate the percentage of customer accounts for which the number of days is between 6 and 48, we will again use the properties of the normal distribution.

Given:

• Mean (\( \mu \)) = 27 days
• Standard deviation (\( \sigma \)) = 7 days

We first calculate the z-scores for the values 6 and 48.

1. For the lower value (6 days): \[ z = \frac{(X - \mu)}{\sigma} = \frac{(6 - 27)}{7} = \frac{-21}{7} = -3 \]

2. For the upper value (48 days): \[ z = \frac{(X - \mu)}{\sigma} = \frac{(48 - 27)}{7} = \frac{21}{7} = 3 \]

Now, we look up these \( z \)-scores in the standard normal distribution:

• A \( z \)-score of -3 corresponds to approximately 0.13% (or about 0.0013 in probability) in the lower tail.
• A \( z \)-score of 3 corresponds to approximately 99.87% (or about 0.9987 in probability) in the upper tail.

To find the percentage of customer accounts within this range, we calculate:

\[ \text{Percentage} = \text{Upper} - \text{Lower} = 99.87% - 0.13% \approx 99.74% \]

Thus, approximately 99.74% of customer accounts have payments made between 6 days and 48 days.