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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?

215215

128128

229229

135135

115115

259259

7676

257257
 
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Part 1

The range of the sample data is 

enter your response here 

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dollars.dollars.

nights squared .nights2.

nights.nights.

dollars squared .dollars2.

​(Round to one decimal place as​ needed.)

To solve the problem, we need to calculate the **range**, **variance**, and **standard deviation** for the given hotel prices. Let's go step by step:

### 1. **List the Data:**
   Prices in dollars: $ 215, 128, 229, 135, 115, 259, 76, 257 $

### 2. **Calculate the Range:**
   The range is the difference between the maximum and minimum values.

   - **Maximum value**: $ 259 $ dollars
   - **Minimum value**: $ 76 $ dollars

   \[
   \text{Range} = \text{Maximum} - \text{Minimum} = 259 - 76 = 183 \text{ dollars}
   \]

### 3. **Calculate the Variance:**
   Variance is calculated using the formula:
   \[
   s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
   \]
   where:
   - $ n $ is the number of observations.
   - $ x_i $ is each individual observation.
   - $ \bar{x} $ is the sample mean.

   **Step 1:** Find the mean ($\bar{x}$):

   \[
   \bar{x} = \frac{215 + 128 + 229 + 135 + 115 + 259 + 76 + 257}{8} = \frac{1414}{8} = 176.75 \text{ dollars}
   \]

   **Step 2:** Calculate each squared deviation from the mean and then the sum of these squared deviations:

   \[
   \text{Sum of squared deviations} = (215 - 176.75)^2 + (128 - 176.75)^2 + \dots + (257 - 176.75)^2
   \]

   Let's calculate each squared deviation:
   - $ (215 - 176.75)^2 = 1476.56 $
   - $ (128 - 176.75)^2 = 2391.06 $
   - $ (229 - 176.75)^2 = 2680.56 $
   - $ (135 - 176.75)^2 = 1761.56 $
   - $ (115 - 176.75)^2 = 3791.06 $
   - $ (259 - 176.75)^2 = 6786.56 $
   - $ (76 - 176.75)^2 = 10158.56 $
   - $ (257 - 176.75)^2 = 6461.56 $

   Sum of squared deviations = $ 141 + 287 + 534 + 259 + 1516 + 1340 + 2378 + 176 = 34196.56 $

   **Step 3:** Divide by $ n-1 $ (since it's a sample):

   \[
   s^2 = \frac{34196.56}{7} \approx 4885.51 \text{ dollars}^2
   \]

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

   \[
   s = \sqrt{4885.51} \approx 69.9 \text{ dollars}
   \]

### 5. **Results with Units:**
- **Range**: $ 183 $ dollars
- **Variance**: $ 4885.51 $ dollars$^2$
- **Standard Deviation**: $ 69.9 $ dollars

### 6. **Interpretation:**
These measures of variation provide useful insights:
- **Range** tells you the price difference between the cheapest and most expensive options, which is quite large ($183). 
- **Standard Deviation** and **Variance** give you an idea of how spread out the prices are around the mean. A higher standard deviation means more price variability, which might indicate that room prices can vary significantly, making it important for someone searching for a room to consider different options carefully.

Would you like more details or have any questions? 

### 5 Relative Questions:
1. How does increasing or decreasing one of the prices affect the standard deviation?
2. What would the variance be if all hotel prices were the same?
3. How do variance and standard deviation differ in terms of their units?
4. Why is it important to use $n-1$ instead of $n$ when calculating the variance for a sample?
5. How does the range compare to the standard deviation in terms of indicating price variability?

### Tip:
Always ensure to include units in your final answers when dealing with measurements, as it helps clarify the meaning of the numerical values.

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?

215, 128, 229, 135, 115, 259, 76, 257

Analyze hotel room prices in a region by calculating the range, variance, and standard deviation. Learn how to apply descriptive statistics to understand price variations, with calculations for an example dataset ranging from $76 to $259. This problem is suitable for high school students in Grades 9-12.

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