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?
162
196
92
137
168
177
264
250
 

To find the range, variance, and standard deviation for the given hotel prices, let's go step by step.

### 1. **Range**:
The range is the difference between the highest and the lowest value in the data set.

Given dataTo find the range, variance, and standard deviation for the given hotel prices, let's go step by step.

### 1. **Range**:
The range is the difference between the highest and the lowest value in the data set.

Given  
\[ 162, 196, 92, 137, 168, 177, 264, 250 \]

- Highest value: \( 264 \)
- Lowest value: \( 92 \)

\[
\text{Range} = 264 - 92 = 172 \, \text{dollars}
\]

### 2. **Variance**:
The variance measures the average of the squared differences from the mean. The formula for sample variance (\( s^2 \)) is:

\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}
\]
where:
- \( x_i \) are the data points,
- \( \bar{x} \) is the sample mean,
- \( n \) is the number of data points.

Let's calculate the mean (\( \bar{x} \)) and the variance.

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

\[
s = \sqrt{s^2}
\]

Let's now calculate these values.### Calculations:

1. **Range**: \( 172 \, \text{dollars} \)
2. **Mean**: \( 180.75 \, \text{dollars} \)
3. **Variance**: \( 3182.5 \, \text{dollars}^2 \)
4. **Standard Deviation**: \( 56.41 \, \text{dollars} \)

### Interpretation:

- The **range** indicates a significant variation between the cheapest and most expensive hotel prices.
- The **standard deviation** shows that most prices deviate by approximately \( 56.41 \) dollars from the mean, suggesting a moderate spread.
- The **variance**, while useful mathematically, is less intuitive for practical use since it is expressed in squared dollars.

For someone searching for a room, these measures highlight the variability in hotel prices, suggesting that room costs can differ significantly depending on the hotel chosen.

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. What would the results look like for a different data set of prices?
2. How would these values change if one extreme value were removed?
3. What is the significance of a low vs. high standard deviation in hotel prices?
4. How does the sample variance differ from population variance?
5. What is the impact of outliers on these measures of variation?

**Tip**: Always be cautious of outliers, as they can significantly affect the variance and standard deviation.

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?
162
196
92
137
168
177
264
250

Learn how to calculate the range, variance, and standard deviation for a set of hotel prices: 162, 196, 92, 137, 168, 177, 264, 250 dollars. This example explains key statistical concepts and provides a step-by-step solution to analyze the variability in hotel prices, perfect for high school students studying statistics.

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