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?

121121

265265

206206

128128

8080

211211

107107

123

To analyze the given data, we will find the range, variance, and standard deviation of the sample.

The given hotel prices are:

- 121, 265, 206, 128, 80, 211, 107, 123

### Step 1: Find the Range

The range is the difference between the highest and lowest values in the data set.

\[
\text{Range} = \text{Max value} - \text{Min value} = 265 - 80 = 185
\]

Thus, the range is **185 dollars**.

### Step 2: Find the Variance

To find the variance, we use the formula for the sample variance:

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

Where:
- $ n $ is the number of data points (in this case, $ n = 8 $)
- $ \bar{x} $ is the sample mean
- $ x_i $ are the individual data points

#### Step 2.1: Calculate the Mean
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{121 + 265 + 206 + 128 + 80 + 211 + 107 + 123}{8} = \frac{1241}{8} = 155.125
\]

#### Step 2.2: Calculate the Squared Differences from the Mean
Now we compute the squared differences for each data point:

\[
\text{For } 121: (121 - 155.125)^2 = 1159.89
\]
\[
\text{For } 265: (265 - 155.125)^2 = 12088.52
\]
\[
\text{For } 206: (206 - 155.125)^2 = 2595.89
\]
\[
\text{For } 128: (128 - 155.125)^2 = 734.89
\]
\[
\text{For } 80: (80 - 155.125)^2 = 5640.89
\]
\[
\text{For } 211: (211 - 155.125)^2 = 3114.39
\]
\[
\text{For } 107: (107 - 155.125)^2 = 2315.64
\]
\[
\text{For } 123: (123 - 155.125)^2 = 1026.89
\]

#### Step 2.3: Sum the Squared Differences and Calculate the Variance
\[
\sum (x_i - \bar{x})^2 = 1159.89 + 12088.52 + 2595.89 + 734.89 + 5640.89 + 3114.39 + 2315.64 + 1026.89 = 28182.01
\]

\[
\text{Variance} = \frac{28182.01}{8 - 1} = \frac{28182.01}{7} = 4026.00
\]

Thus, the variance is **4026.00 dollars$^2$**.

### Step 3: Find the Standard Deviation

The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{4026.00} \approx 63.45
\]

Thus, the standard deviation is **63.45 dollars**.

### Conclusion: Usefulness of the Measures of Variation
- **Range**: It gives a simple idea of how widely the prices vary (from \$80 to \$265), showing the total spread of prices.
- **Variance and Standard Deviation**: These measures show how much prices vary around the mean (\$155.125). A standard deviation of \$63.45 indicates moderate variability, meaning prices are somewhat spread out from the average.
  
For someone searching for a room, the **range** is quite useful as it gives an idea of the lowest and highest prices available. The **standard deviation** helps in understanding how typical prices deviate from the average, which can be helpful for budget planning.

---

Do you want more details on any of these calculations?  
Here are 5 related questions:

1. What are other ways to measure variability in data?
2. How would the variance change if one price was significantly higher?
3. How does the sample variance differ from population variance?
4. What is the impact of outliers on the standard deviation?
5. How can you use the standard deviation to estimate price ranges?

**Tip**: The larger the standard deviation, the more spread out the data points are from the mean.

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?

Learn how to find the range, variance, and standard deviation for a given set of hotel prices. This problem covers key statistical concepts and includes calculations of price variability for someone searching for a hotel room. Suitable for students in Grades 10-12.

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