A survey by a magazine looked at the reliability of cars as the cars get older. They surveyed owners of cars that were 3 years old, and recorded the average yearly repair and maintenance costs (in dollars) for 15 different models of compact cars.
120	55	105	20	100	110	110	50
35	115	50	85	115	100	75	
A button hyperlink to the SALT program that reads: Use SALT.
Find the range (in dollars).
$ 
1130
 
Incorrect: Your answer is incorrect.
Use the range to approximate the value of s (in dollars). (Use 
R = 3s.
 Round your answer to four decimal places.)
$ 
282.5
 
Incorrect: Your answer is incorrect.
Calculate the actual value of s (in dollars). (Round your answer to four decimal places.)

Let's break down this problem step by step to solve it correctly:

### Step 1: Calculate the Range
The **range** of a dataset is the difference between the maximum and minimum values in the data.

Given the dataLet's break down this problem step by step to solve it correctly:

### Step 1: Calculate the Range
The **range** of a dataset is the difference between the maximum and minimum values in the data.

Given the 
\[ 120, 55, 105, 20, 100, 110, 110, 50, 35, 115, 50, 85, 115, 100, 75 \]

1. Find the **maximum value**:
   \[ \text{Max} = 120 \]

2. Find the **minimum value**:
   \[ \text{Min} = 20 \]

3. Calculate the **range**:
   \[
   \text{Range} = \text{Max} - \text{Min} = 120 - 20 = 100
   \]

### Step 2: Use Range to Approximate the Value of $ s $
We are given the approximation formula $ R = 3s $, where $ R $ is the range and $ s $ is the standard deviation. Using the range calculated above:

\[
R = 100 \quad \text{and} \quad R = 3s
\]

To find $ s $, solve for $ s $:

\[
s = \frac{R}{3} = \frac{100}{3} = 33.3333
\]

So, the approximate value of $ s $ is **33.3333 dollars**.

### Step 3: Calculate the Actual Value of $ s $
To calculate the actual value of the standard deviation $ s $, we will use the formula for standard deviation:

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

Where:
- $ n $ is the number of data points (here $ n = 15 $),
- $ x_i $ is each data point,
- $ \bar{x} $ is the mean of the data.

#### 1. Calculate the Mean $ \bar{x} $
The mean is the sum of the data points divided by the number of data points:

\[
\bar{x} = \frac{120 + 55 + 105 + 20 + 100 + 110 + 110 + 50 + 35 + 115 + 50 + 85 + 115 + 100 + 75}{15}
\]

Let me calculate this value:The mean $ \bar{x} $ of the data is 83.

#### 2. Calculate the Standard Deviation $ s $
Now, we can use the standard deviation formula:

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

Let me calculate the standard deviation:The actual standard deviation $ s $ is approximately **32.9935 dollars**.

### Summary:
1. The range of the data is **100 dollars**.
2. The approximate value of $ s $ using $ R = 3s $ is **33.3333 dollars**.
3. The actual value of $ s $ (standard deviation) is **32.9935 dollars**.

Let me know if you'd like more details on any part of the solution!

Here are five related questions:
1. What is the median of the dataset?
2. How would removing the smallest value affect the range?
3. Can the mean and median be the same in all datasets?
4. What is the interquartile range of this dataset?
5. How would increasing each data point by a fixed amount affect the standard deviation?

**Tip:** Standard deviation is sensitive to outliers; a few extreme values can greatly affect its value.

A survey by a magazine looked at the reliability of cars as the cars get older. They surveyed owners of cars that were 3 years old, and recorded the average yearly repair and maintenance costs (in dollars) for 15 different models of compact cars. 120 55 105 20 100 110 110 50 35 115 50 85 115 100 75. Find the range (in dollars). Use the range to approximate the value of s (in dollars). (Use R = 3s. Round your answer to four decimal places.) Calculate the actual value of s (in dollars). (Round your answer to four decimal places.)

Learn how to calculate the range and standard deviation of car repair costs using a sample of 15 compact car models. This problem covers key concepts in descriptive statistics, including the range formula and standard deviation calculations. Suitable for high school students in Grades 10-12.

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