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Let's analyze the problem step by step. We're given hotel prices, and we need to compute the range, variance, and standard deviation for these values.

The prices are:
\[
184, \ 142, \ 185, \ 229, \ 176, \ 260, \ 166, \ 249
\]

### 1. **Range**:
The range is the difference between the maximum and minimum values in the dataset.

\[
\text{Range} = \text{Max} - \text{Min} = 260 - 142 = 118 \text{ dollars}
\]

### 2. **Variance**:
The variance measures how far each number in the set is from the mean (average), and it's computed as follows:

1. Compute the mean (\(\mu\)):

\[
\mu = \frac{184 + 142 + 185 + 229 + 176 + 260 + 166 + 249}{8} = \frac{1591}{8} = 198.875
\]

2. Find the squared deviations from the mean for each price, and then compute the average of those squared deviations:

\[
\text{Variance} = \frac{(184 - 198.875)^2 + (142 - 198.875)^2 + (185 - 198.875)^2 + \dots + (249 - 198.875)^2}{8}
\]

Simplifying each term:

\[
\text{Variance} = \frac{(-14.875)^2 + (-56.875)^2 + (-13.875)^2 + 30.125^2 + (-22.875)^2 + 61.125^2 + (-32.875)^2 + 50.125^2}{8}
\]

\[
\text{Variance} = \frac{221.016 + 3235.766 + 192.516 + 907.531 + 523.766 + 3736.766 + 1080.766 + 2512.531}{8}
\]

\[
\text{Variance} = \frac{12410.658}{8} = 1551.33225
\]

So, the variance is approximately **1551.33 dollars²**.

### 3. **Standard Deviation**:
The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{1551.33225} \approx 39.4 \text{ dollars}
\]

### Interpretation for Someone Searching for a Room:
- **Range**: A room price can vary by up to 118 dollars, which gives an idea of the price fluctuation in the area.
- **Variance and Standard Deviation**: A higher standard deviation (around 39.4 dollars) means there's a moderate spread in prices, so a person might see a variety of pricing options.

Would you like further details on any of these steps, or have questions?

Here are 5 related questions to explore:
1. How would the variance change if the highest or lowest price were removed?
2. How can the standard deviation be used to find outliers in the dataset?
3. What is the significance of variance in comparing two different sets of hotel prices?
4. How does increasing the number of hotel prices affect the range, variance, and standard deviation?
5. Can we assume a normal distribution of prices based on this data?

**Tip**: Understanding how variance and standard deviation work together helps in comparing the price spreads of different datasets.

Listed below are prices in dollars for one night at different hotels in a certain region. Find the range, variance, and standard deviation for the given sample data. Include appropriate units in the results. How useful are the measures of variation for someone searching for a room?

Analyze hotel prices with range, variance, and standard deviation calculations. Learn how price variation affects decision-making when choosing a room. Ideal for Grades 9-12.

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