A sample of size $$ n=10 $$ is drawn from a population. The data is shown below. $$ \begin{array}{llll}108 \quad 88.4 \quad 73 \quad 89.2 \quad 83.3 \ 60.1 & 74.4 \quad 95.1 \quad 108 \quad 108\end{array} $$ What is the range of this data set? range $$ = $$ What is the standard deviation of this data set? (Remember, it is a sample.) Please report the answer with appropriate rounding, reporting 2 more decimal places than the original data. Please, please, please do not calculate the value by hand. stdev $$ = $$

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UpStudy AI · Accepted Answer

Let's analyze the provided data set to determine both the range and the sample standard deviation.

### **Data Set:**
$$
108, \quad 88.4, \quad 73, \quad 89.2, \quad 83.3, \quad 60.1, \quad 74.4, \quad 95.1, \quad 108, \quad 108
$$

### **1. Range of the Data Set**

**Calculation:**
- **Maximum Value:** $$ 108 $$
- **Minimum Value:** $$ 60.1 $$

$$
\text{Range} = \text{Maximum} - \text{Minimum} = 108 - 60.1 = 47.9
$$

$$
\boxed{\text{Range} = 47.9}
$$

### **2. Sample Standard Deviation**

**Step-by-Step Calculation:**

1. **Calculate the Mean ($$ \bar{x} $$):**
$$
\bar{x} = \frac{108 + 88.4 + 73 + 89.2 + 83.3 + 60.1 + 74.4 + 95.1 + 108 + 108}{10} = \frac{887.5}{10} = 88.75
$$

2. **Compute Each Squared Deviation ($$ (x_i - \bar{x})^2 $$):**

| Data Point ($$ x_i $$) | $$ x_i - \bar{x} $$ | $$ (x_i - \bar{x})^2 $$ |
|-----------------------|---------------------|--------------------------|
| 108                   | 19.25               | 370.56                   |
| 88.4                  | -0.35               | 0.12                     |
| 73                    | -15.75              | 248.06                   |
| 89.2                  | 0.45                | 0.20                     |
| 83.3                  | -5.45               | 29.70                    |
| 60.1                  | -28.65              | 820.82                   |
| 74.4                  | -14.35              | 205.92                   |
| 95.1                  | 6.35                | 40.32                    |
| 108                   | 19.25               | 370.56                   |
| 108                   | 19.25               | 370.56                   |
| **Total**             |                     | **2,456.85**             |

3. **Calculate the Sample Variance ($$ s^2 $$):**
$$
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{2456.85}{9} \approx 273.02
$$

4. **Determine the Sample Standard Deviation ($$ s $$):**
$$
s = \sqrt{273.02} \approx 16.52
$$

$$
\boxed{\text{Standard Deviation} = 16.52}
$$

### **Final Answers:**
- **Range:** $$ 47.9 $$
- **Standard Deviation:** $$ 16.52 $$
Range = 47.9 Standard Deviation = 16.52

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