The monthly utility bills in a city are normally​ distributed, with a mean of ​$100 and a standard deviation of ​$16. Find the probability that a randomly selected utility bill is​ (a) less than ​$70​, ​(b) between ​$82 and ​$100​, and​ (c) more than ​$130.

To solve these questions, we use the properties of a normal distribution with mean $\mu = 100$ and standard deviation $\sigma = 16$. The probability calculations are performed using the standard normal distribution (Z-score). For each part, we’ll find the Z-score and use it to determine the probability.

### (a) Probability that a randomly selected utility bill is less than $70

1. **Calculate the Z-score for $70:**
   \[
   Z = \frac{X - \mu}{\sigma} = \frac{70 - 100}{16} = \frac{-30}{16} = -1.875
   \]

2. **Find the probability:**
   Using a standard normal table or calculator, $P(Z < -1.875)$ gives approximately $0.0301$.

   **Answer:** The probability that a utility bill is less than $70 is approximately **0.0301**, or **3.01%**.

### (b) Probability that a utility bill is between $82 and $100

1. **Calculate the Z-score for $82:**
   \[
   Z = \frac{82 - 100}{16} = \frac{-18}{16} = -1.125
   \]

2. **Calculate the Z-score for $100 (mean):**
   \[
   Z = \frac{100 - 100}{16} = 0
   \]

3. **Find the probability:**
   - $P(Z < 0) = 0.5$ (from the mean to the left).
   - $P(Z < -1.125)$ is approximately $0.1303$.

4. **Calculate the difference:**
   \[
   P(-1.125 < Z < 0) = 0.5 - 0.1303 = 0.3697
   \]

   **Answer:** The probability that a utility bill is between $82 and $100 is approximately **0.3697**, or **36.97%**.

### (c) Probability that a utility bill is more than $130

1. **Calculate the Z-score for $130:**
   \[
   Z = \frac{130 - 100}{16} = \frac{30}{16} = 1.875
   \]

2. **Find the probability:**
   - $P(Z > 1.875) = 1 - P(Z < 1.875)$.
   - $P(Z < 1.875)$ is approximately $0.9699$.

3. **Calculate the complement:**
   \[
   P(Z > 1.875) = 1 - 0.9699 = 0.0301
   \]

   **Answer:** The probability that a utility bill is more than $130 is approximately **0.0301**, or **3.01%**.

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Would you like more detailed steps on any part of this solution? Here are some additional questions to consider:

1. How would these probabilities change if the mean utility bill was higher?
2. What is the probability of a bill being between $70 and $130?
3. How can we calculate the probability for bills exceeding one standard deviation above the mean?
4. What is the 90th percentile bill amount in this distribution?
5. If we want a 95% confidence interval for the bill amounts, what would that range be?

**Tip:** For normal distribution problems, remember that Z-scores help standardize any value to assess its relative position within the distribution.

The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $16. Find the probability that a randomly selected utility bill is (a) less than $70, (b) between $82 and $100, and (c) more than $130.

Learn how to calculate probabilities for monthly utility bills, given a normal distribution with a mean of $100 and a standard deviation of $16. This problem covers calculating probabilities for values below, within, and above specific limits, using Z-scores and the standard normal distribution.

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