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?

263

180

113

283

203

119

255

180

Given the hotel prices in dollars for one night:

\[ 263, 180, 113, 283, 203, 119, 255, 180 \]

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

- Minimum value \( \text{min} = 113 \) dollars
- Maximum value \( \text{max} = 283 \) dollars

\[
\text{Range} = \text{max} - \text{min} = 283 - 113 = 170 \text{ dollars}
\]

### Step 2: Find the Variance
To find the variance, we use the formula for sample variance \( s^2 \):

\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
\]

Where:
- \( n = 8 \) (number of data points),
- \( x_i \) represents each data point, and
- \( \bar{x} \) is the sample mean.

#### Calculate the mean \( \bar{x} \):

\[
\bar{x} = \frac{\sum x_i}{n} = \frac{263 + 180 + 113 + 283 + 203 + 119 + 255 + 180}{8} = \frac{1596}{8} = 199.5 \text{ dollars}
\]

#### Calculate each squared deviation \( (x_i - \bar{x})^2 \):

\[
\sum (x_i - \bar{x})^2 = (263 - 199.5)^2 + (180 - 199.5)^2 + \dots + (180 - 199.5)^2 = 4038.25 + 380.25 + \dots + 380.25 = 18738
\]

#### Calculate the sample variance \( s^2 \):

\[
s^2 = \frac{18738}{8-1} = \frac{18738}{7} \approx 2676.86 \text{ dollars}^2
\]

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

\[
s = \sqrt{s^2} \approx \sqrt{2676.86} \approx 51.74 \text{ dollars}
\]

### Interpretation of Results
- **Range**: 170 dollars
- **Variance**: \( \approx 2676.86 \text{ dollars}^2 \)
- **Standard Deviation**: \( \approx 51.74 \text{ dollars} \)

### Usefulness of the Measures of Variation
- **Range**: The range gives a quick sense of the spread between the highest and lowest prices, useful for understanding the price difference across the region.
- **Standard Deviation**: A standard deviation of 51.74 dollars indicates that most hotel prices are within approximately 52 dollars of the average price. This helps in understanding how much variation there is around the average price, which can be useful for budgeting.

For someone searching for a room, these measures provide a good indication of price variability. If prices have a high standard deviation, it suggests a wider variety of price options. A lower standard deviation would indicate that most hotel prices are similar.

Do you want further details or have any questions?

Here are five related questions:
1. How would the standard deviation change if the most expensive hotel price was much higher?
2. Why is it important to square the deviations when calculating variance?
3. How does the variance differ if all hotel prices were identical?
4. What does a large range indicate about the hotel's pricing in a region?
5. How could outliers affect the usefulness of the standard deviation in this context?

**Tip**: When analyzing price data, consider both the range and standard deviation. The range provides the absolute spread, while the standard deviation offers insight into the typical variation from the average price.

Learn how to analyze hotel prices using statistical measures such as range, variance, and standard deviation. Calculate the range, variance, and standard deviation for a sample dataset of hotel prices in dollars. Understand the importance of these measures for someone searching for hotel rooms.

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