Given sample data:x -1 0 1 4f 1 1 3 1The range is ;The sample variance is ;The sample standard deviation is .Round to two decimals.

1. Break Down the Problem

We need to calculate the following statistics based on the given sample data:

1. The range
2. The sample variance
3. The sample standard deviation

2. Relevant Concepts

1. Range: The range is calculated as the difference between the maximum and minimum values in the data set.
2. Sample Variance: The sample variance $s^2$ is calculated using the formula: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$ where $\bar{x}$ is the sample mean and $n$ is the number of observations.
3. Sample Standard Deviation: The sample standard deviation $s$ is simply the square root of the sample variance: $s = \sqrt{s^2}$

3. Analysis and Detail

Sample Data:

$x = [-1, 0, 1, 4, 1, 1, 3, 1]$

1. Calculate the Range:

   • Maximum $= 4$
   • Minimum $= -1$ $\text{Range} = \text{Max} - \text{Min} = 4 - (-1) = 5$

2. Calculate the Sample Mean $\bar{x}$: $\bar{x} = \frac{-1 + 0 + 1 + 4 + 1 + 1 + 3 + 1}{8} = \frac{10}{8} = 1.25$

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

   • For $x_1 = -1$: $( -1 - 1.25 )^2 = (-2.25)^2 = 5.0625$
   • For $x_2 = 0$: $( 0 - 1.25 )^2 = (-1.25)^2 = 1.5625$
   • For $x_3 = 1$: $( 1 - 1.25 )^2 = (-0.25)^2 = 0.0625$
   • For $x_4 = 4$: $( 4 - 1.25 )^2 = (2.75)^2 = 7.5625$
   • For $x_5 = 1$: $( 1 - 1.25 )^2 = (-0.25)^2 = 0.0625$
   • For $x_6 = 1$: $( 1 - 1.25 )^2 = (-0.25)^2 = 0.0625$
   • For $x_7 = 3$: $( 3 - 1.25 )^2 = (1.75)^2 = 3.0625$
   • For $x_8 = 1$: $( 1 - 1.25 )^2 = (-0.25)^2 = 0.0625$

   Summing these values: $\sum (x_i - \bar{x})^2 = 5.0625 + 1.5625 + 0.0625 + 7.5625 + 0.0625 + 0.0625 + 3.0625 + 0.0625 = 17.5$

   Now substituting back into the variance formula: $s^2 = \frac{17.5}{8 - 1} = \frac{17.5}{7} \approx 2.5$

4. Calculate the Sample Standard Deviation $s$: $s = \sqrt{s^2} = \sqrt{2.5} \approx 1.58$

4. Verify and Summarize

• Range: 5
• Sample Variance: 2.5
• Sample Standard Deviation: 1.58

Final Answer

• The range is 5.
• The sample variance is 2.5.
• The sample standard deviation is 1.58.