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?

100100

9494

294294

160160

9191

140140

119119

167167
 
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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.)

Let's start by calculating the range, variance, and standard deviation for the given sample data. The values provided are:

100, 94, 294, 160, 91, 140, 119, 167

### Part 1: Calculate the Range

The range is calculated by subtracting the smallest value from the largest value in the dataset:

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

Here, the maximum value is 294, and the minimum value is 91. So:

\[
\text{Range} = 294 - 91 = 203 \, \text{dollars}
\]

### Part 2: Variance and Standard Deviation

1. **Step 1**: Find the mean (average):
   \[
   \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
   \]

   Let's calculate that first.

   The sum of all values:
   \[
   100 + 94 + 294 + 160 + 91 + 140 + 119 + 167 = 1165
   \]

   The number of values is 8, so the mean is:
   \[
   \text{Mean} = \frac{1165}{8} = 145.625
   \]

2. **Step 2**: Calculate the squared differences from the mean for each value.

   For each value $x$, the squared difference is $ (x - \text{mean})^2 $. Let's calculate these:

   - For 100: $ (100 - 145.625)^2 = 2088.89 $
   - For 94: $ (94 - 145.625)^2 = 2675.64 $
   - For 294: $ (294 - 145.625)^2 = 21920.64 $
   - For 160: $ (160 - 145.625)^2 = 204.39 $
   - For 91: $ (91 - 145.625)^2 = 2980.89 $
   - For 140: $ (140 - 145.625)^2 = 31.64 $
   - For 119: $ (119 - 145.625)^2 = 716.39 $
   - For 167: $ (167 - 145.625)^2 = 448.39 $

3. **Step 3**: Find the variance.

   The variance is the average of the squared differences. Since this is a sample, we divide by $n-1 = 8-1 = 7$.

   \[
   \text{Variance} = \frac{2088.89 + 2675.64 + 21920.64 + 204.39 + 2980.89 + 31.64 + 716.39 + 448.39}{7} = 4480.41 \, \text{dollars}^2
   \]

4. **Step 4**: Find the standard deviation.

   The standard deviation is the square root of the variance:
   \[
   \text{Standard deviation} = \sqrt{4480.41} = 66.96 \, \text{dollars}
   \]

### Summary of Results:

- **Range**: 203 dollars
- **Variance**: 4480.41 dollars²
- **Standard Deviation**: 66.96 dollars

### How useful are these measures of variation?

The range gives a sense of the spread between the highest and lowest prices, which is useful for quickly assessing how much variation exists in hotel prices. The standard deviation provides a more nuanced understanding of the average variation around the mean, showing that the prices tend to vary by about $67 from the average. The variance is less directly useful but is part of the calculation of standard deviation. 

These measures can help a person decide whether they can expect large price differences and whether the average price is a reliable indicator of the room prices available.

---

Would you like further explanation or have any questions? 

Here are 5 questions you could explore next:
1. How does the distribution of prices affect standard deviation?
2. Why is it important to use $n-1$ for sample variance?
3. What other statistical measures could help a traveler choose a hotel?
4. How would removing outliers like the highest price affect variance?
5. Can we calculate a confidence interval for the average hotel price?

**Tip**: Understanding the standard deviation helps you see how much variability to expect in data, which is crucial for decision-making.

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? 100, 94, 294, 160, 91, 140, 119, 167.

Explore how to calculate the range, variance, and standard deviation for hotel prices in a region. This step-by-step solution walks you through key statistical concepts, with a focus on how variation in room prices can help travelers make informed decisions. Suitable for Grades 9-12.

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