Monthly utility bills in a certain city are normally distributed with a mean of $121 and a standard deviation of $15. What is the cutoff amount for the top 10% of the bills? Use the normal table and SHOW ALL WORK to answer this
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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![### Example Problem: Calculating the Top 10% Cutoff for Utility Bills
**Problem Statement:**
Monthly utility bills in a certain city are normally distributed with a mean of $121 and a standard deviation of $15. What is the cutoff amount for the top 10% of the bills?
**Solution:** To determine the cutoff amount for the top 10% of the utility bills, we'll use the properties of the normal distribution and the standard normal (Z) table.
#### Step-by-Step Solution:
1. **Identify the Mean and Standard Deviation:**
- Mean (μ): $121
- Standard Deviation (σ): $15
2. **Determine the Z-score that corresponds to the top 10%:**
The top 10% refers to the 90th percentile (since 100% - 10% = 90%). We need to find the Z-score that corresponds to the 90th percentile in the standard normal distribution.
3. **Using the Standard Normal Table:**
- Look for the value in the Z-table that is closest to 0.9000.
- The value corresponding to 0.9000 is approximately 1.28.
4. **Apply the Z-score Formula:**
The Z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]
Where:
- **Z** is the Z-score,
- **X** is the cutoff amount,
- **μ** is the mean,
- **σ** is the standard deviation.
Rearranging the formula to solve for **X** gives us:
\[
X = Z \cdot \sigma + \mu
\]
5. **Calculate the Cutoff Amount:**
\[
X = 1.28 \cdot 15 + 121
\]
\[
X = 19.2 + 121
\]
\[
X = 140.2
\]
**Conclusion:**
The cutoff amount for the top 10% of the utility bills is $140.20.
### Visualization:
**Normal Distribution Curve:**
Imagine a bell-shaped curve where the center (mean) is at $121. The area under the curve to the right of the cutoff point (approximately $140.20) covers 10% of the total area.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa35d32dc-632b-40fd-aa41-e2c0040899b1%2Fa260cf74-9d7a-4afe-8329-eb05fb58fa21%2Fe5t9i9h_processed.png&w=3840&q=75)
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