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Part 1
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?
166
167
255
272
289
169
278
268
 
Question content area bottom
Part 1
The range of the sample data is 
  
enter your response here 
▼ 
nights.
dollars.
dollars squared .
nights squared .
​(Round to one decimal place as​ needed.)

To solve this problem, we will find the **range**, **variance**, and **standard deviation** for the sample data provided.

The data set for hotel prices is:  
166, 167, 255, 272, 289, 169, 278, 268.

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

\[
\text{Range} = \text{Maximum value} - \text{Minimum value}
\]

For the given dataTo solve this problem, we will find the **range**, **variance**, and **standard deviation** for the sample data provided.

The data set for hotel prices is:  
166, 167, 255, 272, 289, 169, 278, 268.

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

\[
\text{Range} = \text{Maximum value} - \text{Minimum value}
\]

For the given 
\[
\text{Maximum value} = 289, \quad \text{Minimum value} = 166
\]
\[
\text{Range} = 289 - 166 = 123
\]

Thus, the range is **123 dollars**.

### 2. **Variance**:
Variance measures the spread of the data points. The formula for the sample variance $ s^2 $ is:

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

Where:
- $ n $ is the sample size,
- $ x_i $ are the individual data points, and
- $ \bar{x} $ is the sample mean.

#### Step 1: Calculate the mean $ \bar{x} $
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{166 + 167 + 255 + 272 + 289 + 169 + 278 + 268}{8} = 233
\]

#### Step 2: Find the squared differences from the mean for each data point:
\[
(166 - 233)^2 = 4489, \quad (167 - 233)^2 = 4356, \quad (255 - 233)^2 = 484, \quad (272 - 233)^2 = 1521,
\]
\[
(289 - 233)^2 = 3136, \quad (169 - 233)^2 = 4096, \quad (278 - 233)^2 = 2025, \quad (268 - 233)^2 = 1225
\]

#### Step 3: Calculate the sum of these squared differences:
\[
4489 + 4356 + 484 + 1521 + 3136 + 4096 + 2025 + 1225 = 21284
\]

#### Step 4: Divide by $ n-1 = 8-1 = 7 $:
\[
s^2 = \frac{21284}{7} = 3040.6
\]

Thus, the **variance** is approximately **3040.6 dollars squared**.

### 3. **Standard Deviation**:
The standard deviation is the square root of the variance:
\[
s = \sqrt{3040.6} \approx 55.2
\]

Thus, the **standard deviation** is approximately **55.2 dollars**.

### Summary of results:
- **Range** = 123 dollars
- **Variance** = 3040.6 dollars squared
- **Standard Deviation** = 55.2 dollars

### Interpretation:
The **range** gives an idea of the spread between the cheapest and most expensive hotel. The **variance** and **standard deviation** tell us how much the prices fluctuate around the mean price. These measures are useful for someone searching for a room because they indicate the variability in prices, helping the person decide whether to look for a cheaper or more expensive room.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the variance change if a new price of $200 were added to the data set?
2. How does the range help in making hotel pricing comparisons?
3. Can standard deviation alone be used to find outliers in this data?
4. What does it mean if the standard deviation were significantly larger?
5. How would the results differ if the sample were doubled in size?

**Tip**: In many practical scenarios, the standard deviation is preferred over variance, since it has the same units as the original data.

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? 166, 167, 255, 272, 289, 169, 278, 268

Discover how to calculate the range, variance, and standard deviation for a sample of hotel prices, including values like $166, $167, and $289. This tutorial walks you through the formulas and provides insights on how these statistical measures are useful in analyzing hotel price variations. Suitable for Grades 10-12.

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