Question
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 25 with a standard deviation of 4 days. Assume the data to be approximately bell-shaped.
a) Estimate the percentage of bills for which payment was made in greater than 33 days.
a) Estimate the percentage of bills for which payment was made in greater than 33 days.
Answers
GPT-4o mini
To solve this problem, we can use the properties of a normal distribution since the data is assumed to be approximately bell-shaped (normally distributed).
Given:
- Mean (μ) = 25 days
- Standard deviation (σ) = 4 days
We want to find the percentage of bills for which payment was made in greater than 33 days.
1. First, we need to compute the z-score for 33 days. The z-score is calculated using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we are interested in (33 days),
- \(\mu\) is the mean (25 days),
- \(\sigma\) is the standard deviation (4 days).
Substituting in the values:
\[
z = \frac{(33 - 25)}{4} = \frac{8}{4} = 2
\]
2. Now, we can use the z-score to find the corresponding percentage in the standard normal distribution table (or by using a calculator or software that provides cumulative probabilities).
The z-score of 2 corresponds to the cumulative probability of about 0.9772 (or 97.72%). This means that approximately 97.72% of the payments are made in 33 days or less.
3. To find the percentage of bills for which payment was made in **greater** than 33 days, we subtract this value from 100%:
\[
P(X > 33) = 1 - P(X < 33)
\]
\[
P(X > 33) = 1 - 0.9772 = 0.0228
\]
4. Finally, convert this probability to a percentage:
\[
P(X > 33) \times 100 = 0.0228 \times 100 = 2.28\%
\]
So, approximately **2.28%** of the bills are paid in greater than 33 days.
Given:
- Mean (μ) = 25 days
- Standard deviation (σ) = 4 days
We want to find the percentage of bills for which payment was made in greater than 33 days.
1. First, we need to compute the z-score for 33 days. The z-score is calculated using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we are interested in (33 days),
- \(\mu\) is the mean (25 days),
- \(\sigma\) is the standard deviation (4 days).
Substituting in the values:
\[
z = \frac{(33 - 25)}{4} = \frac{8}{4} = 2
\]
2. Now, we can use the z-score to find the corresponding percentage in the standard normal distribution table (or by using a calculator or software that provides cumulative probabilities).
The z-score of 2 corresponds to the cumulative probability of about 0.9772 (or 97.72%). This means that approximately 97.72% of the payments are made in 33 days or less.
3. To find the percentage of bills for which payment was made in **greater** than 33 days, we subtract this value from 100%:
\[
P(X > 33) = 1 - P(X < 33)
\]
\[
P(X > 33) = 1 - 0.9772 = 0.0228
\]
4. Finally, convert this probability to a percentage:
\[
P(X > 33) \times 100 = 0.0228 \times 100 = 2.28\%
\]
So, approximately **2.28%** of the bills are paid in greater than 33 days.
(b) Estimate the percentage of bills for which payment was made in less than 21 days.
GPT-4o mini
To find the percentage of bills for which payment was made in less than 21 days, we will again use the z-score formula and the properties of the normal distribution.
Given:
- Mean (μ) = 25 days
- Standard deviation (σ) = 4 days
We want to find the z-score for 21 days.
1. **Calculate the z-score** using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) = 21 days,
- \(\mu\) = 25 days,
- \(\sigma\) = 4 days.
Substituting the values into the formula:
\[
z = \frac{(21 - 25)}{4} = \frac{-4}{4} = -1
\]
2. **Find the cumulative probability associated with the z-score** of -1. Using the standard normal distribution table or a calculator, we can find that:
The cumulative probability for \(z = -1\) is approximately \(0.1587\). This means that about 15.87% of bills were paid in less than 21 days.
3. **Convert to percentage:**
\[
P(X < 21) \approx 0.1587 \times 100 = 15.87\%
\]
Thus, approximately **15.87%** of the bills are paid in less than 21 days.
Given:
- Mean (μ) = 25 days
- Standard deviation (σ) = 4 days
We want to find the z-score for 21 days.
1. **Calculate the z-score** using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) = 21 days,
- \(\mu\) = 25 days,
- \(\sigma\) = 4 days.
Substituting the values into the formula:
\[
z = \frac{(21 - 25)}{4} = \frac{-4}{4} = -1
\]
2. **Find the cumulative probability associated with the z-score** of -1. Using the standard normal distribution table or a calculator, we can find that:
The cumulative probability for \(z = -1\) is approximately \(0.1587\). This means that about 15.87% of bills were paid in less than 21 days.
3. **Convert to percentage:**
\[
P(X < 21) \approx 0.1587 \times 100 = 15.87\%
\]
Thus, approximately **15.87%** of the bills are paid in less than 21 days.
(c) Estimate the percentage of bills for which payment was made between 17 and 25 days.
GPT-4o mini
To find the percentage of bills for which payment was made between 17 and 25 days, we will calculate the z-scores for both 17 days and 25 days, and then find the cumulative probabilities associated with these z-scores.
### Step 1: Calculate the z-scores
Given:
- Mean (μ) = 25 days
- Standard deviation (σ) = 4 days
**For 17 days:**
\[
z_{17} = \frac{(17 - 25)}{4} = \frac{-8}{4} = -2
\]
**For 25 days:**
\[
z_{25} = \frac{(25 - 25)}{4} = \frac{0}{4} = 0
\]
### Step 2: Find cumulative probabilities
Using the standard normal distribution table or appropriate calculator:
1. **Cumulative probability for \( z = -2\):**
- \( P(Z < -2) \approx 0.0228 \) (about 2.28%).
2. **Cumulative probability for \( z = 0\):**
- \( P(Z < 0) = 0.5 \) (since 0 is the mean, the area to the left of z = 0 is 50%).
### Step 3: Find the probability of payment between 17 and 25 days
To find the probability of payments made between 17 and 25 days, we subtract the cumulative probability of \(z = -2\) from that of \(z = 0\):
\[
P(17 < X < 25) = P(Z < 0) - P(Z < -2)
\]
\[
P(17 < X < 25) = 0.5 - 0.0228 = 0.4772
\]
### Step 4: Convert the probability to a percentage
\[
P(17 < X < 25) \times 100 = 0.4772 \times 100 = 47.72\%
\]
Thus, approximately **47.72%** of the bills are paid between 17 and 25 days.
### Step 1: Calculate the z-scores
Given:
- Mean (μ) = 25 days
- Standard deviation (σ) = 4 days
**For 17 days:**
\[
z_{17} = \frac{(17 - 25)}{4} = \frac{-8}{4} = -2
\]
**For 25 days:**
\[
z_{25} = \frac{(25 - 25)}{4} = \frac{0}{4} = 0
\]
### Step 2: Find cumulative probabilities
Using the standard normal distribution table or appropriate calculator:
1. **Cumulative probability for \( z = -2\):**
- \( P(Z < -2) \approx 0.0228 \) (about 2.28%).
2. **Cumulative probability for \( z = 0\):**
- \( P(Z < 0) = 0.5 \) (since 0 is the mean, the area to the left of z = 0 is 50%).
### Step 3: Find the probability of payment between 17 and 25 days
To find the probability of payments made between 17 and 25 days, we subtract the cumulative probability of \(z = -2\) from that of \(z = 0\):
\[
P(17 < X < 25) = P(Z < 0) - P(Z < -2)
\]
\[
P(17 < X < 25) = 0.5 - 0.0228 = 0.4772
\]
### Step 4: Convert the probability to a percentage
\[
P(17 < X < 25) \times 100 = 0.4772 \times 100 = 47.72\%
\]
Thus, approximately **47.72%** of the bills are paid between 17 and 25 days.